# Prerequisites

## Factoring Polynomials

### Learning Objectives:

1. Master a proven technique for note taking: The Cornell Method.
2. Use the Cornell Process to study from your notes.

### Objective 1: Master a proven technique for note taking: The Cornell Method.

The Cornell Method for taking notes was created by an education professor at Cornell University, Dr. Walter Pauk. The Cornell Method consists of two strategies, a format for notetaking and then the process of using your notes to study. The format for note taking involves separating out the pages of your notebook into three or four separate regions. This method can help summarize information from a lecture, a video, or a reading from your text. You don’t need to purchase Cornell paper, just divide your sheets into regions like the illustration below. Write on just one side of the paper so that you can later fold back the left column and quiz yourself using practice test questions.

The top portion is called the heading. This is where you write your name, the class, the date, the section of the text and a main topic objective for the day.

The right section is where you will take notes during class. Try to summarize main ideas here without copying word for word everything your teacher is saying. Use bullet points or numbers to prioritize important ideas. Include here the definitions your teacher presents, and the examples worked in class. It is useful if you can group the content here by learning objectives.
The left column about 2 inches wide is used to write questions about the main concepts that were covered in class. For example, write sample test questions over the concepts discussed in class in this column. This is also where you should write cues about the importance of the information including vocabulary terms, diagrams, formulas or the methods being used to solve. If you like to sketch this is a place to add illustrations for key ideas or icons so you can quickly identify certain components.

The summary will be written in the lower section about 2 inches from the bottom. You will complete this after class summarizing the important concepts you were taught in a short compact way. Think of this summary as what you might describe to a friend who happened to miss class that day. Reflect on the important ideas here. Add illustrations or a mind map to your summary if this helps. You will use this section to find information later when studying.

### Practice Makes Perfect

### Objective 2: Use the Cornell Process to study from your notes.

The Cornell Way. Using your Cornell notes to study is referred to as the Cornell Way.
The process includes Cornell note taking (presented in Objective 1), note making, note interacting, and note reflecting.
After taking notes in class, follow up with these study methods.

Note making: This is where you will fill in any gaps left in your Cornell notes meaning add in any missed details. Fill in your question column with sample test questions, any formulas used, and highlight, circle or star important any important ideas. Complete your summary row with a few sentences summarizing the important ideas. This should be completed right after class or within the next day. Be creative and add some color, make these notes into something you enjoy working with.

Note interacting: This is the ongoing process of studying from your notes. Fold your left question column back and ask yourself the practice test questions. Include note interactions in the review for your exams. These note interactions can be as short as 5 minutes in length but need to happen regularly and at least daily for the week before exams.

Note reflecting: This is where you assess how helpful your notes were. Do this right after you get back your graded exam. Did the regular note interactions help you to perform better on the exam? Were there problems similar to those you predicted on the exam? Enhance your notes with any important ideas you had initially left out. Then use what you learn from this assessment to improve your note taking in the future.

### Practice Makes Perfect

Imagine that we are trying to find the area of a lawn so that we can determine how much grass seed to purchase. The lawn is the green portion in .
The area of the entire region can be found using the formula for the area of a rectangle.
The areas of the portions that do not require grass seed need to be subtracted from the area of the entire region. The two square regions each have an area of
  units2. The other rectangular region has one side of length
  and one side of length
  giving an area of
  units2. So the region that must be subtracted has an area of
  units2.
The area of the region that requires grass seed is found by subtracting
  units2. This area can also be expressed in factored form as
  units2. We can confirm that this is an equivalent expression by multiplying.
Many polynomial expressions can be written in simpler forms by factoring. In this section, we will look at a variety of methods that can be used to factor polynomial expressions.

### Factoring the Greatest Common Factor of a Polynomial

When we study fractions, we learn that the greatest common factor (GCF) of two numbers is the largest number that divides evenly into both numbers. For instance,
  is the GCF of
  and
  because it is the largest number that divides evenly into both
  and
  The GCF of polynomials works the same way:
  is the GCF of
  and
  because it is the largest polynomial that divides evenly into both
  and

When factoring a polynomial expression, our first step should be to check for a GCF. Look for the GCF of the coefficients, and then look for the GCF of the variables.

### Factoring a Trinomial with Leading Coefficient 1

Although we should always begin by looking for a GCF, pulling out the GCF is not the only way that polynomial expressions can be factored. The polynomial
  has a GCF of 1, but it can be written as the product of the factors
  and

Trinomials of the form
  can be factored by finding two numbers with a product of
  and a sum of
  The trinomial
  for example, can be factored using the numbers
  and
  because the product of those numbers is
  and their sum is
  The trinomial can be rewritten as the product of
  and

### Factoring by Grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial
  can be rewritten as
  using this process. We begin by rewriting the original expression as
  and then factor each portion of the expression to obtain
  We then pull out the GCF of
  to find the factored expression.

###  Factoring a Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be written as the square of a binomial. Recall that when a binomial is squared, the result is the square of the first term added to twice the product of the two terms and the square of the last term.
We can use this equation to factor any perfect square trinomial.

### Factoring a Difference of Squares

A difference of squares is a perfect square subtracted from a perfect square. Recall that a difference of squares can be rewritten as factors containing the same terms but opposite signs because the middle terms cancel each other out when the two factors are multiplied.
We can use this equation to factor any differences of squares.

### Factoring the Sum and Difference of Cubes

Now, we will look at two new special products: the sum and difference of cubes. Although the sum of squares cannot be factored, the sum of cubes can be factored into a binomial and a trinomial.
Similarly, the difference of cubes can be factored into a binomial and a trinomial, but with different signs.
We can use the acronym SOAP to remember the signs when factoring the sum or difference of cubes. The first letter of each word relates to the signs: Same Opposite Always Positive. For example, consider the following example.
The sign of the first 2 is the same as the sign between
  The sign of the
  term is opposite the sign between
  And the sign of the last term, 4, is always positive.

### Factoring Expressions with Fractional or Negative Exponents

Expressions with fractional or negative exponents can be factored by pulling out a GCF. Look for the variable or exponent that is common to each term of the expression and pull out that variable or exponent raised to the lowest power. These expressions follow the same factoring rules as those with integer exponents. For instance,
  can be factored by pulling out
  and being rewritten as

### Key Equations

1. The greatest common factor, or GCF, can be factored out of a polynomial. Checking for a GCF should be the first step in any factoring problem. See .
2. Trinomials with leading coefficient 1 can be factored by finding numbers that have a product of the third term and a sum of the second term. See .
3. Trinomials can be factored using a process called factoring by grouping. See .
4. Perfect square trinomials and the difference of squares are special products and can be factored using equations. See  and .
5. The sum of cubes and the difference of cubes can be factored using equations. See  and .
6. Polynomials containing fractional and negative exponents can be factored by pulling out a GCF. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the greatest common factor.
For the following exercises, factor by grouping.
For the following exercises, factor the polynomial.
For the following exercises, factor the polynomials.

### Real-World Applications

For the following exercises, consider this scenario:
Charlotte has appointed a chairperson to lead a city beautification project. The first act is to install statues and fountains in one of the city’s parks. The park is a rectangle with an area of
  m2, as shown in the figure below. The length and width of the park are perfect factors of the area.
For the following exercise, consider the following scenario:
A school is installing a flagpole in the central plaza. The plaza is a square with side length 100 yd. as shown in the figure below.  The flagpole will take up a square plot with area
  yd2.

### Extensions

For the following exercises, factor the polynomials completely.


# Prerequisites

## Rational Expressions

### Learning Objectives

1. Identify the skills leading to successful preparation for a college level mathematics exam.

2. Create a plan for success when taking mathematics exams.

### Objective 1: Identify the skills leading to successful preparation for a college level mathematics exam.

Complete the following surveys by placing a checkmark in the a column for each strategy based on the frequency that you engaged in the strategy during your last academic term.

### Practice Makes Perfect

Practice: Identify the skills leading to successful preparation for a college level mathematics exam.

### Objective 2: Create a plan for success when taking mathematics exams.

1. It’s important to take the opportunity to reflect on your past experiences in taking math exams as you begin a new term. We can learn a lot from these reflections and thus work toward developing a strategy for improvement.
In the table below list 5 challenges you have had in past math courses when taking an exam and list a possible solution that you could try this semester.

2. Develop your plan for success.  Keep in mind the idea of mindsets and try to approach your test taking strategies with a growth mindset. Now is the time for growth as you begin a new term. Share your plan with your study group members.

A pastry shop has fixed costs of
  per week and variable costs of
  per box of pastries. The shop’s costs per week in terms of
  the number of boxes made, is
  We can divide the costs per week by the number of boxes made to determine the cost per box of pastries.
Notice that the result is a polynomial expression divided by a second polynomial expression. In this section, we will explore quotients of polynomial expressions.

### Simplifying Rational Expressions

The quotient of two polynomial expressions is called a rational expression. We can apply the properties of fractions to rational expressions, such as simplifying the expressions by canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let’s start with the rational expression shown.
We can factor the numerator and denominator to rewrite the expression.
Then we can simplify that expression by canceling the common factor

### Multiplying Rational Expressions

Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.

### Dividing Rational Expressions

Division of rational expressions works the same way as division of other fractions. To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. Using this approach, we would rewrite
  as the product
  Once the division expression has been rewritten as a multiplication expression, we can multiply as we did before.

### Adding and Subtracting Rational Expressions

Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.
We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.
The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were
  and
  then the LCD would be

Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of
  by
  and the expression with a denominator of
  by

### Simplifying Complex Rational Expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression
  can be simplified by rewriting the numerator as the fraction
  and combining the expressions in the denominator as
  We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get
  which is equal to

### Key Concepts

1. Rational expressions can be simplified by cancelling common factors in the numerator and denominator. See .
2. We can multiply rational expressions by multiplying the numerators and multiplying the denominators. See .
3. To divide rational expressions, multiply by the reciprocal of the second expression. See .
4. Adding or subtracting rational expressions requires finding a common denominator. See  and .
5. Complex rational expressions have fractions in the numerator or the denominator. These expressions can be simplified. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, simplify the rational expressions.
For the following exercises, multiply the rational expressions and express the product in simplest form.
For the following exercises, divide the rational expressions.
For the following exercises, add and subtract the rational expressions, and then simplify.
For the following exercises, simplify the rational expression.

### Real-World Applications

### Extensions

For the following exercises, perform the given operations and simplify.

### Chapter Review Exercises

### Real Numbers: Algebra Essentials

For the following exercises, perform the given operations.
For the following exercises, solve the equation.
For the following exercises, simplify the expression.
For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.

### Exponents and Scientific Notation

For the following exercises, simplify the expression.

### Radicals and Rational Expressions

For the following exercises, find the principal square root.

### Polynomials

For the following exercises, perform the given operations and simplify.

### Factoring Polynomials

For the following exercises, find the greatest common factor.
For the following exercises, factor the polynomial.

### Rational Expressions

For the following exercises, simplify the expression.

### Chapter Practice Test

For the following exercises, identify the number as rational, irrational, whole, or natural. Choose the most descriptive answer.
For the following exercises, evaluate the expression.
For the following exercises, simplify the expression.
For the following exercises, factor the polynomial.
For the following exercises, simplify the expression.


# Equations and Inequalities

## Introduction to Equations and Inequalities

Irrigation is a critical aspect of agriculture, which can expand the yield of farms and enable farming in areas not naturally viable for crops. But the materials, equipment, and the water itself are expensive and complex. To be efficient and productive, farm owners and irrigation specialists must carefully lay out the network of pipes, pumps, and related equipment. The available land can be divided into regular portions (similar to a grid), and the different sizes of irrigation systems and conduits can be installed within the plotted area.


# Equations and Inequalities

## The Rectangular Coordinate Systems and Graphs

### Learning Objectives

1. Plot points on a real number line (IA 1.4.7)
2. Plot points in a rectangular coordinate system (IA 3.1.1)

### Objective 1:  Plot points on a real number line (IA 1.4.7)

### Practice Makes Perfect

Plot each set of points on a real number line.

### Objective 2: Plot points in a rectangular coordinate system (IA 3.1.1)

### Practice Makes Perfect

Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.
Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in . Laying a rectangular coordinate grid over the map, we can see that each stop aligns with an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in locations.

### Plotting Ordered Pairs in the Cartesian Coordinate System

An old story describes how seventeenth-century philosopher/mathematician René Descartes, while sick in bed, invented the system that has become the foundation of algebra. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly’s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that it was possible to locate any object in a two-dimensional plane using just two numbers—the displacement from the horizontal axis and the displacement from the vertical axis.
While there is evidence that ideas similar to Descartes’ grid system existed centuries earlier, it was Descartes who introduced the components that comprise the Cartesian coordinate system, a grid system having perpendicular axes. Descartes named the horizontal axis the  and the vertical axis the .
The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the x-axis and the y-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a quadrant; the quadrants are numbered counterclockwise as shown in
The center of the plane is the point at which the two axes cross. It is known as the origin, or point
  From the origin, each axis is further divided into equal units: increasing, positive numbers to the right on the x-axis and up the y-axis; decreasing, negative numbers to the left on the x-axis and down the y-axis. The axes extend to positive and negative infinity as shown by the arrowheads in .
Each point in the plane is identified by its , or horizontal displacement from the origin, and its , or vertical displacement from the origin. Together, we write them as an ordered pair indicating the combined distance from the origin in the form
  An ordered pair is also known as a coordinate pair because it consists of x- and y-coordinates. For example, we can represent the point
  in the plane by moving three units to the right of the origin in the horizontal direction, and one unit down in the vertical direction. See .
When dividing the axes into equally spaced increments, note that the x-axis may be considered separately from the y-axis. In other words, while the x-axis may be divided and labeled according to consecutive integers, the y-axis may be divided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against the balance in a savings account, or quantity against cost, and so on. Consider the rectangular coordinate system primarily as a method for showing the relationship between two quantities.

### Graphing Equations by Plotting Points

We can plot a set of points to represent an equation. When such an equation contains both an x variable and a y variable, it is called an equation in two variables. Its graph is called a graph in two variables. Any graph on a two-dimensional plane is a graph in two variables.
Suppose we want to graph the equation
  We can begin by substituting a value for x into the equation and determining the resulting value of y. Each pair of x- and y-values is an ordered pair that can be plotted.  lists values of x from –3 to 3 and the resulting values for y.
We can plot the points in the table. The points for this particular equation form a line, so we can connect them. See . This is not true for all equations.
Note that the x-values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of x to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.

### Graphing Equations with a Graphing Utility

Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated so they are written in the style
  The TI-84 Plus, and many other calculator makes and models, have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of a graph can be seen.
For example, the equation
  has been entered in the TI-84 Plus shown in a. In b, the resulting graph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen on the TI-84 Plus shows
  and
  See c.
By changing the window to show more of the positive x-axis and more of the negative y-axis, we have a much better view of the graph and the x- and y-intercepts. See a and b.

### Finding x-intercepts and y-intercepts

The intercepts of a graph are points at which the graph crosses the axes. The  is the point at which the graph crosses the x-axis. At this point, the y-coordinate is zero. The  is the point at which the graph crosses the y-axis. At this point, the x-coordinate is zero.
To determine the x-intercept, we set y equal to zero and solve for x. Similarly, to determine the y-intercept, we set x equal to zero and solve for y. For example, lets find the intercepts of the equation

To find the x-intercept, set

To find the y-intercept, set

We can confirm that our results make sense by observing a graph of the equation as in . Notice that the graph crosses the axes where we predicted it would.

### Using the Distance Formula

Derived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem,
  is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hypotenuse. See .
The relationship of sides
  and
  to side d is the same as that of sides a and b to side c. We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example,
  ) The symbols
  and
  indicate that the lengths of the sides of the triangle are positive. To find the length c, take the square root of both sides of the Pythagorean Theorem.
It follows that the distance formula is given as
We do not have to use the absolute value symbols in this definition because any number squared is positive.

### Using the Midpoint Formula

When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the midpoint formula. Given the endpoints of a line segment,
  and
  the midpoint formula states how to find the coordinates of the midpoint

A graphical view of a midpoint is shown in . Notice that the line segments on either side of the midpoint are congruent.

### Key Concepts

1. We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as displacement from the x-axis and displacement from the y-axis. See .
2. An equation can be graphed in the plane by creating a table of values and plotting points. See .
3. Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations usually have to be entered in the form y=_____. See .
4. Finding the x- and y-intercepts can define the graph of a line. These are the points where the graph crosses the axes. See .
5. The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment. See  and .
6. The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the x-coordinates and the sum of the y-coordinates of the endpoints by 2. See  and .

### Section Exercises

### Verbal

### Algebraic

For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.
For each of the following exercises, solve the equation for y in terms of x.
For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.
For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

### Graphical

For each of the following exercises, identify the information requested.
For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).
For each of the following exercises, construct a table and graph the equation by plotting at least three points.

### Numeric

For each of the following exercises, find and plot the x- and y-intercepts, and graph the straight line based on those two points.
For each of the following exercises, use the graph in the figure below.

### Technology

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.
After graphing it, use the 2nd CALC button and 1:value button, hit enter. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0, thus finding the y-intercept, for each of the following graphs.
For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.
After graphing it, use the 2nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.
Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

### Extensions

### Real-World Applications


# Equations and Inequalities

## Linear Equations in One Variable

### Learning Objectives

1. Simplify expressions using order of operations (IA 1.1.3)
2. Solve linear equations using a general strategy (IA 2.1.1)

### Objective 1: Simplify expressions using order of operations (IA 1.1.3)

### Practice Makes Perfect

Evaluate the following expressions being sure to follow the order of operations:
Simplify by combining like terms:

### Objective 2: Solve linear equations using a general strategy (IA 2.1.1)

### Practice Makes Perfect

Solve each linear equation using the general strategy.
Caroline is a full-time college student planning a spring break vacation. To earn enough money for the trip, she has taken a part-time job at the local bank that pays $15.00/hr, and she opened a savings account with an initial deposit of $400 on January 15. She arranged for direct deposit of her payroll checks. If spring break begins March 20 and the trip will cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation? If she can only work 4 hours per day, how many days per week will she have to work? How many weeks will it take? In this section, we will investigate problems like this and others, which generate graphs like the line in .

### Solving Linear Equations in One Variable

A linear equation is an equation of a straight line, written in one variable. The only power of the variable is 1. Linear equations in one variable may take the form
  and are solved using basic algebraic operations.
We begin by classifying linear equations in one variable as one of three types: identity, conditional, or inconsistent. An identity equation is true for all values of the variable. Here is an example of an identity equation.
The solution set consists of all values that make the equation true. For this equation, the solution set is all real numbers because any real number substituted for
  will make the equation true.
A conditional equation is true for only some values of the variable. For example, if we are to solve the equation
  we have the following:
The solution set consists of one number:
  It is the only solution and, therefore, we have solved a conditional equation.
An inconsistent equation results in a false statement. For example, if we are to solve
  we have the following:
Indeed,
  There is no solution because this is an inconsistent equation.
Solving linear equations in one variable involves the fundamental properties of equality and basic algebraic operations. A brief review of those operations follows.

### Solving a Rational Equation

In this section, we look at rational equations that, after some manipulation, result in a linear equation. If an equation contains at least one rational expression, it is a considered a rational equation.
Recall that a rational number is the ratio of two numbers, such as
  or
  A rational expression is the ratio, or quotient, of two polynomials. Here are three examples.
Rational equations have a variable in the denominator in at least one of the terms.
Our goal is to perform algebraic operations so that the variables appear in the numerator. In fact, we will eliminate all denominators by multiplying both sides of the equation by the least common denominator (LCD).
Finding the LCD is identifying an expression that contains the highest power of all of the factors in all of the denominators. We do this because when the equation is multiplied by the LCD, the common factors in the LCD and in each denominator will equal one and will cancel out.
A common mistake made when solving rational equations involves finding the LCD when one of the denominators is a binomial—two terms added or subtracted—such as
  Always consider a binomial as an individual factor—the terms cannot be separated. For example, suppose a problem has three terms and the denominators are

  and
  First, factor all denominators. We then have

  and
  as the denominators. (Note the parentheses placed around the second denominator.) Only the last two denominators have a common factor of
  The
  in the first denominator is separate from the
  in the
  denominators. An effective way to remember this is to write factored and binomial denominators in parentheses, and consider each parentheses as a separate unit or a separate factor. The LCD in this instance is found by multiplying together the
  one factor of
  and the 3. Thus, the LCD is the following:
So, both sides of the equation would be multiplied by
  Leave the LCD in factored form, as this makes it easier to see how each denominator in the problem cancels out.
Another example is a problem with two denominators, such as
  and
  Once the second denominator is factored as
  there is a common factor of x in both denominators and the LCD is

Sometimes we have a rational equation in the form of a proportion; that is, when one fraction equals another fraction and there are no other terms in the equation.
We can use another method of solving the equation without finding the LCD: cross-multiplication. We multiply terms by crossing over the equal sign.
Multiply
  and
  which results in

Any solution that makes a denominator in the original expression equal zero must be excluded from the possibilities.

### Finding a Linear Equation

Perhaps the most familiar form of a linear equation is the slope-intercept form, written as
  where
  and
  Let us begin with the slope.

### The Slope of a Line

The slope of a line refers to the ratio of the vertical change in y over the horizontal change in x between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.
If the slope is positive, the line slants to the right. If the slope is negative, the line slants to the left. As the slope increases, the line becomes steeper. Some examples are shown in . The lines indicate the following slopes:

  and

### The Point-Slope Formula

Given the slope and one point on a line, we can find the equation of the line using the point-slope formula.
This is an important formula, as it will be used in other areas of college algebra and often in calculus to find the equation of a tangent line. We need only one point and the slope of the line to use the formula. After substituting the slope and the coordinates of one point into the formula, we simplify it and write it in slope-intercept form.

### Standard Form of a Line

Another way that we can represent the equation of a line is in standard form. Standard form is given as
where

  and
  are integers. The x- and y-terms are on one side of the equal sign and the constant term is on the other side.

### Vertical and Horizontal Lines

The equations of vertical and horizontal lines do not require any of the preceding formulas, although we can use the formulas to prove that the equations are correct. The equation of a vertical line is given as
where c is a constant. The slope of a vertical line is undefined, and regardless of the y-value of any point on the line, the x-coordinate of the point will be c.
Suppose that we want to find the equation of a line containing the following points:
  and
  First, we will find the slope.
Zero in the denominator means that the slope is undefined and, therefore, we cannot use the point-slope formula. However, we can plot the points. Notice that all of the x-coordinates are the same and we find a vertical line through
  See .
The equation of a horizontal line is given as
where c is a constant. The slope of a horizontal line is zero, and for any x-value of a point on the line, the y-coordinate will be c.
Suppose we want to find the equation of a line that contains the following set of points:
  and
  We can use the point-slope formula. First, we find the slope using any two points on the line.
Use any point for
  in the formula, or use the y-intercept.
The graph is a horizontal line through
  Notice that all of the y-coordinates are the same. See .

### Determining Whether Graphs of Lines are Parallel or Perpendicular

Parallel lines have the same slope and different y-intercepts. Lines that are parallel to each other will never intersect. For example,  shows the graphs of various lines with the same slope,

All of the lines shown in the graph are parallel because they have the same slope and different y-intercepts.
Lines that are perpendicular intersect to form a
  -angle. The slope of one line is the negative reciprocal of the other. We can show that two lines are perpendicular if the product of the two slopes is
  For example,  shows the graph of two perpendicular lines. One line has a slope of 3; the other line has a slope of

### Writing the Equations of Lines Parallel or Perpendicular to a Given Line

As we have learned, determining whether two lines are parallel or perpendicular is a matter of finding the slopes. To write the equation of a line parallel or perpendicular to another line, we follow the same principles as we do for finding the equation of any line. After finding the slope, use the point-slope formula to write the equation of the new line.

### Key Concepts

1. We can solve linear equations in one variable in the form
  using standard algebraic properties. See  and .
2. A rational expression is a quotient of two polynomials. We use the LCD to clear the fractions from an equation. See  and .
3. All solutions to a rational equation should be verified within the original equation to avoid an undefined term, or zero in the denominator. See  and  and .
4. Given two points, we can find the slope of a line using the slope formula. See .
5. We can identify the slope and y-intercept of an equation in slope-intercept form. See .
6. We can find the equation of a line given the slope and a point. See .
7. We can also find the equation of a line given two points. Find the slope and use the point-slope formula. See .
8. The standard form of a line has no fractions. See .
9. Horizontal lines have a slope of zero and are defined as
  where c is a constant.
10. Vertical lines have an undefined slope (zero in the denominator), and are defined as
  where c is a constant. See .
11. Parallel lines have the same slope and different y-intercepts. See  and .
12. Perpendicular lines have slopes that are negative reciprocals of each other unless one is horizontal and the other is vertical. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, solve the equation for

For the following exercises, solve each rational equation for
  State all x-values that are excluded from the solution set.
For the following exercises, find the equation of the line using the point-slope formula.
Write all the final equations using the slope-intercept form.
For the following exercises, find the equation of the line using the given information.

### Graphical

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither.

### Numeric

For the following exercises, find the slope of the line that passes through the given points.
For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular.

### Technology

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as Y1, then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values.

### Extensions

### Real-World Applications

For the following exercises, use this scenario: The cost of renting a car is $45/wk plus $0.25/mi traveled during that week. An equation to represent the cost would be
  where
  is the number of miles traveled.


# Equations and Inequalities

## Models and Applications

### Learning Objectives

1. Solve a formula for a specified variable (IA 2.3.1)
2. Use a problem-solving strategy for word problems (IA 2.2.1)

### Objective 1: Solve a formula for a specified variable (IA 2.3.1)

### Practice Makes Perfect

Solve each formula for the specific variable.

### Objective 2: Use a problem-solving strategy for word problems (IA 2.2.1)

### Practice Makes Perfect

Use a Problem-Solving Strategy for word problems.
Neka is hoping to get an A in his college algebra class. He has scores of 75, 82, 95, 91, and 94 on his first five tests. Only the final exam remains, and the maximum number of points that can be earned is 100. Is it possible for Neka to end the course with an A? A simple linear equation will give Neka his answer.
Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to produce x widgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.

### Setting up a Linear Equation to Solve a Real-World Application

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as $0.10/mi, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write
  This expression represents a variable cost because it changes according to the number of miles driven.
If a quantity is independent of a variable, we usually just add or subtract it, according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges $0.10/mi plus a daily fee of $50. We can use these quantities to model an equation that can be used to find the daily car rental cost

When dealing with real-world applications, there are certain expressions that we can translate directly into math.  lists some common verbal expressions and their equivalent mathematical expressions.

### Using a Formula to Solve a Real-World Application

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region,
  the perimeter of a rectangle,
  and the volume of a rectangular solid,
  When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

### Key Concepts

1. A linear equation can be used to solve for an unknown in a number problem. See .
2. Applications can be written as mathematical problems by identifying known quantities and assigning a variable to unknown quantities. See .
3. There are many known formulas that can be used to solve applications. Distance problems, for example, are solved using the
  formula. See .
4. Many geometry problems are solved using the perimeter formula
  the area formula
  or the volume formula
  See , , and .

### Section Exercises

### Verbal

### Real-World Applications

For the following exercises, use the information to find a linear algebraic equation model to use to answer the question being asked.
For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of $20 and charges of $.05/min for calls. Company B has a monthly fee of $5 and charges $.10/min for calls.
For the following exercises, use this scenario: A wireless carrier offers the following plans that a person is considering. The Family Plan: $90 monthly fee, unlimited talk and text on up to 8 lines, and data charges of $40 for each device for up to 2 GB of data per device. The Mobile Share Plan: $120 monthly fee for up to 10 devices, unlimited talk and text for all the lines, and data charges of $35 for each device up to a shared total of 10 GB of data. Use
  for the number of devices that need data plans as part of their cost.
For exercises 17 and 18, use this scenario: A retired woman has $50,000 to invest but needs to make $6,000 a year from the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%.
For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $75/wk plus $.10/mi driven. Plan B charges $100/wk plus $.05/mi driven.
For the following exercises, use the formula given to solve for the required value.
For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.


# Equations and Inequalities

## Complex Numbers

### Learning Objectives

1. Use the product property to simplify radical expressions (IA 8.2.1)
2. Evaluate the square root of a negative number (IA 8.8.1)

### Objective 1: Use the product property to simplify radical expressions (IA 8.2.1)

### Practice Makes Perfect

Simplify a radical expression using the Product Property.

### Objective 2: Evaluate the square root of a negative number (IA 8.8.1)

Imaginary numbers result when we try to take the square root of a negative number. They do not belong to the set of real numbers and so are called imaginary or complex. We will see complex solutions when we solve quadratic equations whose graph does not touch the x-axis. Imaginary numbers are used in many real-life applications including the study of electricity involving alternating current (AC) electronics. Wireless, cellular and radar technologies utilize imaginary numbers. Electrical engineers use imaginary numbers to measure the amplitude and phase of electrical oscillations.

### Practice Makes Perfect

Evaluate the square root of a negative number. Perform any indicated operations and simplify.
Discovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The image is built on the theory of self-similarity and the operation of iteration. Zooming in on a fractal image brings many surprises, particularly in the high level of repetition of detail that appears as magnification increases. The equation that generates this image turns out to be rather simple.
In order to better understand it, we need to become familiar with a new set of numbers. Keep in mind that the study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set of numbers that fills voids in the set of real numbers and find out how to work within it.

### Expressing Square Roots of Negative Numbers as Multiples of

We know how to find the square root of any positive real number. In a similar way, we can find the square root of any negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number. The imaginary number
  is defined as the square root of

So, using properties of radicals,
We can write the square root of any negative number as a multiple of
  Consider the square root of

We use
  and not
  because the principal root of
  is the positive root.
A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written
  where
  is the real part and
  is the imaginary part. For example,
  is a complex number. So, too, is

Imaginary numbers differ from real numbers in that a squared imaginary number produces a negative real number. Recall that when a positive real number is squared, the result is a positive real number and when a negative real number is squared, the result is also a positive real number. Complex numbers consist of real and imaginary numbers.

### Plotting a Complex Number on the Complex Plane

We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number, we need to address the two components of the number. We use the complex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs
  where
  represents the coordinate for the horizontal axis and
  represents the coordinate for the vertical axis.
Let’s consider the number
  The real part of the complex number is
  and the imaginary part is 3. We plot the ordered pair
  to represent the complex number
  as shown in .

### Adding and Subtracting Complex Numbers

Just as with real numbers, we can perform arithmetic operations on complex numbers. To add or subtract complex numbers, we combine the real parts and then combine the imaginary parts.

### Multiplying Complex Numbers

Multiplying complex numbers is much like multiplying binomials. The major difference is that we work with the real and imaginary parts separately.

### Multiplying a Complex Number by a Real Number

Lets begin by multiplying a complex number by a real number. We distribute the real number just as we would with a binomial. Consider, for example,
  :

### Multiplying Complex Numbers Together

Now, let’s multiply two complex numbers. We can use either the distributive property or more specifically the FOIL method because we are dealing with binomials. Recall that FOIL is an acronym for multiplying First, Inner, Outer, and Last terms together. The difference with complex numbers is that when we get a squared term,
  it equals

### Dividing Complex Numbers

Dividing two complex numbers is more complicated than adding, subtracting, or multiplying because we cannot divide by an imaginary number, meaning that any fraction must have a real-number denominator to write the answer in standard form
  We need to find a term by which we can multiply the numerator and the denominator that will eliminate the imaginary portion of the denominator so that we end up with a real number as the denominator. This term is called the complex conjugate of the denominator, which is found by changing the sign of the imaginary part of the complex number. In other words, the complex conjugate of
  is
  For example, the product of
  and
  is
The result is a real number.
Note that complex conjugates have an opposite relationship: The complex conjugate of
  is
  and the complex conjugate of
  is
  Further, when a quadratic equation with real coefficients has complex solutions, the solutions are always complex conjugates of one another.
Suppose we want to divide
  by
  where neither
  nor
  equals zero. We first write the division as a fraction, then find the complex conjugate of the denominator, and multiply.
Multiply the numerator and denominator by the complex conjugate of the denominator.
Apply the distributive property.
Simplify, remembering that

### Simplifying Powers of i

The powers of
  are cyclic. Let’s look at what happens when we raise
  to increasing powers.
We can see that when we get to the fifth power of
  it is equal to the first power. As we continue to multiply
  by increasing powers, we will see a cycle of four. Let’s examine the next four powers of

The cycle is repeated continuously:
  every four powers.

### Key Concepts

1. The square root of any negative number can be written as a multiple of
  See .
2. To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. See .
3. Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts. See .
4. Complex numbers can be multiplied and divided.

5. The powers of
  are cyclic, repeating every fourth one. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, evaluate the algebraic expressions.

### Graphical

For the following exercises, plot the complex numbers on the complex plane.

### Numeric

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

### Technology

For the following exercises, use a calculator to help answer the questions.

### Extensions

For the following exercises, evaluate the expressions, writing the result as a simplified complex number.


# Equations and Inequalities

## Quadratic Equations

### Learning Objectives

1. Recognize and use the appropriate method to factor a polynomial completely (IA 6.4.1)

### Objective 1: Recognize and use the appropriate method to factor a polynomial completely  (IA 6.4.1)

The following chart summarizes the factoring methods and outlines a strategy you should use when factoring polynomials.

### Practice Makes Perfect

Recognize and use the appropriate method to factor a polynomial completely. Factor each of the following polynomials completely, if a polynomial does not factor label it as prime.

The computer monitor on the left in  is a 23.6-inch model and the one on the right is a 27-inch model. Proportionally, the monitors appear very similar. If there is a limited amount of space and we desire the largest monitor possible, how do we decide which one to choose? In this section, we will learn how to solve problems such as this using four different methods.

### Solving Quadratic Equations by Factoring

An equation containing a second-degree polynomial is called a quadratic equation. For example, equations such as
  and
  are quadratic equations. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.
Often the easiest method of solving a quadratic equation is factoring. Factoring means finding expressions that can be multiplied together to give the expression on one side of the equation.
If a quadratic equation can be factored, it is written as a product of linear terms. Solving by factoring depends on the zero-product property, which states that if
  then
  or
  where a and b are real numbers or algebraic expressions. In other words, if the product of two numbers or two expressions equals zero, then one of the numbers or one of the expressions must equal zero because zero multiplied by anything equals zero.
Multiplying the factors expands the equation to a string of terms separated by plus or minus signs. So, in that sense, the operation of multiplication undoes the operation of factoring. For example, expand the factored expression
  by multiplying the two factors together.
The product is a quadratic expression. Set equal to zero,
  is a quadratic equation. If we were to factor the equation, we would get back the factors we multiplied.
The process of factoring a quadratic equation depends on the leading coefficient, whether it is 1 or another integer. We will look at both situations; but first, we want to confirm that the equation is written in standard form,
  where a, b, and c are real numbers, and
  The equation
  is in standard form.
We can use the zero-product property to solve quadratic equations in which we first have to factor out the greatest common factor (GCF), and for equations that have special factoring formulas as well, such as the difference of squares, both of which we will see later in this section.

### Solving Quadratics with a Leading Coefficient of 1

In the quadratic equation
  the leading coefficient, or the coefficient of
  is 1. We have one method of factoring quadratic equations in this form.

### Solving a Quadratic Equation by Factoring when the Leading Coefficient is not 1

When the leading coefficient is not 1, we factor a quadratic equation using the method called grouping, which requires four terms. With the equation in standard form, let’s review the grouping procedures:
1. With the quadratic in standard form,
  multiply

2. Find two numbers whose product equals
  and whose sum equals

3. Rewrite the equation replacing the
  term with two terms using the numbers found in step 1 as coefficients of x.
4. Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping.
5. Factor out the expression in parentheses.
6. Set the expressions equal to zero and solve for the variable.

### Using the Square Root Property

When there is no linear term in the equation, another method of solving a quadratic equation is by using the square root property, in which we isolate the
  term and take the square root of the number on the other side of the equals sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the
  term so that the square root property can be used.

### Completing the Square

Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use a method for solving a quadratic equation known as completing the square. Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, a, must equal 1. If it does not, then divide the entire equation by a. Then, we can use the following procedures to solve a quadratic equation by completing the square.
We will use the example
  to illustrate each step.
1. Given a quadratic equation that cannot be factored, and with

2. Multiply the

3.
Add

4. The left side of the equation can now be factored as a perfect square.

5. Use the square root property and solve.

6. The solutions are

### Using the Quadratic Formula

The fourth method of solving a quadratic equation is by using the quadratic formula, a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.
We can derive the quadratic formula by completing the square. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by
  and obtain a positive a. Given

  we will complete the square as follows:
1. First, move the constant term to the right side of the equal sign:

2. As we want the leading coefficient to equal 1, divide through by

3. Then, find

4. Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:

5. Now, use the square root property, which gives

6. Finally, add

### The Discriminant

The quadratic formula not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the discriminant, or the expression under the radical,
  The discriminant tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect.  relates the value of the discriminant to the solutions of a quadratic equation.

### Using the Pythagorean Theorem

One of the most famous formulas in mathematics is the Pythagorean Theorem. It is based on a right triangle, and states the relationship among the lengths of the sides as
  where
  and
  refer to the legs of a right triangle adjacent to the
  angle, and
  refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications.
We use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other two. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for the missing side.
The Pythagorean Theorem is given as
where
  and
  refer to the legs of a right triangle adjacent to the
  angle, and
  refers to the hypotenuse, as shown in .

### Key Equations

### Key Concepts

1. Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the equation is a difference of squares. The zero-product property is then used to find solutions. See , , and .
2. Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. See  and .
3. Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. See  and .
4. Completing the square is a method of solving quadratic equations when the equation cannot be factored. See .
5. A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See  and .
6. The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. See .
7. The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, solve the quadratic equation by factoring.
For the following exercises, solve the quadratic equation by using the square root property.
For the following exercises, solve the quadratic equation by completing the square. Show each step.
For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.
For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

### Technology

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

### Extensions

### Real-World Applications


# Equations and Inequalities

## Other Types of Equations

We have solved linear equations, rational equations, and quadratic equations using several methods. However, there are many other types of equations, and we will investigate a few more types in this section. We will look at equations involving rational exponents, polynomial equations, radical equations, absolute value equations, equations in quadratic form, and some rational equations that can be transformed into quadratics. Solving any equation, however, employs the same basic algebraic rules. We will learn some new techniques as they apply to certain equations, but the algebra never changes.

### Solving Equations Involving Rational Exponents

Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example,
  is another way of writing

  is another way of writing
  The ability to work with rational exponents is a useful skill, as it is highly applicable in calculus.
We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals 1. For example,

  and so on.

### Solving Equations Using Factoring

We have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial equations, which are equations that contain a string of terms including numerical coefficients and variables. When we are faced with an equation containing polynomials of degree higher than 2, we can often solve them by factoring.

### Solving Radical Equations

Radical equations are equations that contain variables in the radicand (the expression under a radical symbol), such as
Radical equations may have one or more radical terms, and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations, as it is not unusual to find extraneous solutions, roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. However, checking each answer in the original equation will confirm the true solutions.

### Solving an Absolute Value Equation

Next, we will learn how to solve an absolute value equation. To solve an equation such as
  we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is
  or
  This leads to two different equations we can solve independently.
Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

### Solving Other Types of Equations

There are many other types of equations in addition to the ones we have discussed so far. We will see more of them throughout the text. Here, we will discuss equations that are in quadratic form, and rational equations that result in a quadratic.

### Solving Equations in Quadratic Form

Equations in quadratic form are equations with three terms. The first term has a power other than 2. The middle term has an exponent that is one-half the exponent of the leading term. The third term is a constant. We can solve equations in this form as if they were quadratic. A few examples of these equations include
  and
  In each one, doubling the exponent of the middle term equals the exponent on the leading term. We can solve these equations by substituting a variable for the middle term.

### Solving Rational Equations Resulting in a Quadratic

Earlier, we solved rational equations. Sometimes, solving a rational equation results in a quadratic. When this happens, we continue the solution by simplifying the quadratic equation by one of the methods we have seen. It may turn out that there is no solution.

### Key Concepts

1. Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. See , , and .
2. Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. See  and .
3. We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index. See  and .
4. To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value. See .
5. Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant. We may also see a binomial in place of the single variable. We use substitution to solve. See  and .
6. Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, solve the rational exponent equation. Use factoring where necessary.
For the following exercises, solve the following polynomial equations by grouping and factoring.
For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.
For the following exercises, solve the equation involving absolute value.
For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.

### Extensions

For the following exercises, solve for the unknown variable.

### Real-World Applications

For the following exercises, use the model for the period of a pendulum,
  such that
  where the length of the pendulum is L and the acceleration due to gravity is

For the following exercises, use a model for body surface area, BSA, such that
  where w = weight in kg and h = height in cm.


# Equations and Inequalities

## Linear Inequalities and Absolute Value Inequalities

It is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.

### Using Interval Notation

Indicating the solution to an inequality such as
    can be achieved in several ways.
We can use a number line as shown in . The blue ray begins at
    and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.
We can use set-builder notation:
    which translates to “all real numbers x such that x is greater than or equal to 4.” Notice that braces are used to indicate a set.
The third method is interval notation, in which solution sets are indicated with parentheses or brackets. The solutions to
    are represented as
    This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.
The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval, or a set of numbers in which a solution falls, are
    or all numbers between
    and
    including
    but not including

    all real numbers between, but not including
    and
    and
    all real numbers less than and including
     outlines the possibilities.

### Using the Properties of Inequalities

When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.

### Solving Inequalities in One Variable Algebraically

As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.

### Understanding Compound Inequalities

A compound inequality includes two inequalities in one statement. A statement such as
    means
    and
    There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.

### Solving Absolute Value Inequalities

As we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at
    has an absolute value of
    as it is x units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.
An absolute value inequality is an equation of the form
Where A, and sometimes B, represents an algebraic expression dependent on a variable x. Solving the inequality means finding the set of all
    -values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.
There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.
Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of x-values such that the distance between
    and 600 is less than or equal to 200. We represent the distance between
    and 600 as
    and therefore,
    or
This means our returns would be between $400 and $800.
To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.

### Key Concepts

1. Interval notation is a method to indicate the solution set to an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well. See  and .
2. Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality. See , , , and .
3. Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values, which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities. See  and .
4. Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value. See  and .
5. Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by graphing, as we cannot depend on a visual for a precise solution. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, solve the inequality. Write your final answer in interval notation.
For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation.
For the following exercises, describe all the x-values within or including a distance of the given values.
For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation.

### Graphical

For the following exercises, graph the function. Observe the points of intersection and shade the x-axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation.
For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being y2) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the y-values of the lines.

### Numeric

For the following exercises, write the set in interval notation.
For the following exercises, write the interval in set-builder notation.
For the following exercises, write the set of numbers represented on the number line in interval notation.

### Technology

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter y2 = the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, 1:abs(. Find the points of intersection, recall (2nd CALC 5:intersection, 1st curve, enter, 2nd curve, enter, guess, enter). Copy a sketch of the graph and shade the x-axis for your solution set to the inequality. Write final answers in interval notation.

### Extensions

### Real-World Applications

### Chapter Review Exercises

### The Rectangular Coordinate Systems and Graphs

For the following exercises, find the x-intercept and the y-intercept without graphing.
For the following exercises, solve for y in terms of x, putting the equation in slope–intercept form.
For the following exercises, find the distance between the two points.
For the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.
For the following exercises, construct a table and graph the equation by plotting at least three points.

### Linear Equations in One Variable

For the following exercises, solve for

For the following exercises, solve for
    State all x-values that are excluded from the solution set.
For the following exercises, find the equation of the line using the point-slope formula.

### Models and Applications

For the following exercises, write and solve an equation to answer each question.

### Complex Numbers

For the following exercises, use the quadratic equation to solve.
For the following exercises, name the horizontal component and the vertical component.
For the following exercises, perform the operations indicated.

### Quadratic Equations

For the following exercises, solve the quadratic equation by factoring.
For the following exercises, solve the quadratic equation by using the square-root property.
For the following exercises, solve the quadratic equation by completing the square.
For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No real solution.
For the following exercises, solve the quadratic equation by the method of your choice.

### Other Types of Equations

For the following exercises, solve the equations.

### Linear Inequalities and Absolute Value Inequalities

For the following exercises, solve the inequality. Write your final answer in interval notation.
For the following exercises, solve the compound inequality. Write your answer in interval notation.
For the following exercises, graph as described.

### Chapter Practice Test

For the following exercises, find the equation of the line with the given information.
For the following exercises, find the real solutions of each equation by factoring.


# Functions

## Introduction to Functions

Toward the end of the twentieth century, the values of stocks of Internet and technology companies rose dramatically. As a result, the Standard and Poor’s stock market average rose as well. The graph above tracks the value of that initial investment of just under $100 over the 40 years. It shows that an investment that was worth less than $500 until about 1995 skyrocketed up to about $1100 by the beginning of 2000. That five-year period became known as the “dot-com bubble” because so many Internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventually burst. Many companies grew too fast and then suddenly went out of business. The result caused the sharp decline represented on the graph beginning at the end of 2000.
Notice, as we consider this example, that there is a definite relationship between the year and stock market average. For any year we choose, we can determine the corresponding value of the stock market average. In this chapter, we will explore these kinds of relationships and their properties.


# Functions

## Functions and Function Notation

### Learning Objectives

1. Find the value of a function (IA 3.5.3)

### Objective 1: Find the value of a function (IA 3.5.3)

A relation is any set of ordered pairs, (x,y). The collection of x-values in the ordered pairs together make up the domain. The collection of y-values in the ordered pairs together make up the range.
A special type of relation, called a function, is studied extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.

### Practice Makes Perfect

Find the value of a function
A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

### Determining Whether a Relation Represents a Function

A relation is a set of ordered pairs. The set of the first components of each ordered pair is called the domain and the set of the second components of each ordered pair is called the range. Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.
The domain is

 The range is

Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter
  Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter

A function
  is a relation that assigns a single value in the range to each value in the domain. In other words, no x-values are repeated. For our example that relates the first five natural numbers to numbers double their values, this relation is a function because each element in the domain,
  is paired with exactly one element in the range,

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

Notice that each element in the domain,

 is not paired with exactly one element in the range,

For example, the term “odd” corresponds to three values from the range,

 and the term “even” corresponds to two values from the range,

 This violates the definition of a function, so this relation is not a function.
 compares relations that are functions and not functions.

### Using Function Notation

Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions. A standard function notation is one representation that facilitates working with functions.
To represent “height is a function of age,” we start by identifying the descriptive variables

 for height and

 for age. The letters
  and
  are often used to represent functions just as we use
  and
  to represent numbers and

 and
  to represent sets.

Remember, we can use any letter to name the function; the notation
 shows us that
 depends on
  The value
 must be put into the function
 to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.
We can also give an algebraic expression as the input to a function. For example
 means “first add a and b, and the result is the input for the function f.” The operations must be performed in this order to obtain the correct result.

### Representing Functions Using Tables

A common method of representing functions is in the form of a table. The table rows or columns display the corresponding input and output values. In some cases, these values represent all we know about the relationship; other times, the table provides a few select examples from a more complete relationship.
 lists the input number of each month (January = 1, February = 2, and so on) and the output value of the number of days in that month. This information represents all we know about the months and days for a given year (that is not a leap year). Note that, in this table, we define a days-in-a-month function
 where
 identifies months by an integer rather than by name.
 defines a function
 Remember, this notation tells us that
 is the name of the function that takes the input
 and gives the output

 displays the age of children in years and their corresponding heights. This table displays just some of the data available for the heights and ages of children. We can see right away that this table does not represent a function because the same input value, 5 years, has two different output values, 40 in. and 42 in.

### Finding Input and Output Values of a Function

When we know an input value and want to determine the corresponding output value for a function, we evaluate the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.
When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function’s formula and solve for the input. Solving can produce more than one solution because different input values can produce the same output value.

### Evaluation of Functions in Algebraic Forms

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example, the function
 can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.

### Evaluating Functions Expressed in Formulas

Some functions are defined by mathematical rules or procedures expressed in equation form. If it is possible to express the function output with a formula involving the input quantity, then we can define a function in algebraic form. For example, the equation
 expresses a functional relationship between

and
  We can rewrite it to decide if
  is a function of

### Evaluating a Function Given in Tabular Form

As we saw above, we can represent functions in tables. Conversely, we can use information in tables to write functions, and we can evaluate functions using the tables. For example, how well do our pets recall the fond memories we share with them? There is an urban legend that a goldfish has a memory of 3 seconds, but this is just a myth. Goldfish can remember up to 3 months, while the beta fish has a memory of up to 5 months. And while a puppy’s memory span is no longer than 30 seconds, the adult dog can remember for 5 minutes. This is meager compared to a cat, whose memory span lasts for 16 hours.
The function that relates the type of pet to the duration of its memory span is more easily visualized with the use of a table. See .http://www.kgbanswers.com/how-long-is-a-dogs-memory-span/4221590. Accessed 3/24/2014.
At times, evaluating a function in table form may be more useful than using equations.Here let us call the function

 The domain of the function is the type of pet and the range is a real number representing the number of hours the pet’s memory span lasts. We can evaluate the function
 at the input value of “goldfish.” We would write
  Notice that, to evaluate the function in table form, we identify the input value and the corresponding output value from the pertinent row of the table. The tabular form for function
 seems ideally suited to this function, more so than writing it in paragraph or function form.

### Finding Function Values from a Graph

Evaluating a function using a graph also requires finding the corresponding output value for a given input value, only in this case, we find the output value by looking at the graph. Solving a function equation using a graph requires finding all instances of the given output value on the graph and observing the corresponding input value(s).

### Determining Whether a Function is One-to-One

Some functions have a given output value that corresponds to two or more input values. For example, in the stock chart shown in the figure at the beginning of this chapter, the stock price was $1000 on five different dates, meaning that there were five different input values that all resulted in the same output value of $1000.
However, some functions have only one input value for each output value, as well as having only one output for each input. We call these functions one-to-one functions. As an example, consider a school that uses only letter grades and decimal equivalents, as listed in .
This grading system represents a one-to-one function, because each letter input yields one particular grade point average output and each grade point average corresponds to one input letter.
To visualize this concept, let’s look again at the two simple functions sketched in (a) and (b). The function in part (a) shows a relationship that is not a one-to-one function because inputs
  and
  both give output
  The function in part (b) shows a relationship that is a one-to-one function because each input is associated with a single output.

### Using the Vertical Line Test

As we have seen in some examples above, we can represent a function using a graph. Graphs display a great many input-output pairs in a small space. The visual information they provide often makes relationships easier to understand. By convention, graphs are typically constructed with the input values along the horizontal axis and the output values along the vertical axis.
The most common graphs name the input value
  and the output value
  and we say
  is a function of
  or
  when the function is named
  The graph of the function is the set of all points
  in the plane that satisfies the equation
  If the function is defined for only a few input values, then the graph of the function is only a few points, where the x-coordinate of each point is an input value and the y-coordinate of each point is the corresponding output value. For example, the black dots on the graph in  tell us that
  and
  However, the set of all points
  satisfying
  is a curve. The curve shown includes
  and
  because the curve passes through those points.
The vertical line test can be used to determine whether a graph represents a function. If we can draw any vertical line that intersects a graph more than once, then the graph does not define a function because a function has only one output value for each input value. See .

###  Using the Horizontal Line Test

Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph. If any horizontal line intersects the graph more than once, then the graph does not represent a one-to-one function.

### Identifying Basic Toolkit Functions

In this text, we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a base set of building-block elements. We call these our “toolkit functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use
  as the input variable and
  as the output variable.
We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in .

### Key Equations

### Key Concepts

1. A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output. See  and .
2. Function notation is a shorthand method for relating the input to the output in the form
 See  and .
3. In tabular form, a function can be represented by rows or columns that relate to input and output values. See .
4. To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. See  and .
5. To solve for a specific function value, we determine the input values that yield the specific output value. See .
6. An algebraic form of a function can be written from an equation. See  and .
7. Input and output values of a function can be identified from a table. See .
8. Relating input values to output values on a graph is another way to evaluate a function. See .
9. A function is one-to-one if each output value corresponds to only one input value. See .
10. A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See .
11. The graph of a one-to-one function passes the horizontal line test. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the relation represents a function.
For the following exercises, determine whether the relation represents
 as a function of

For the following exercises, evaluate

### Graphical

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.
For the following exercises, determine if the given graph is a one-to-one function.

### Numeric

For the following exercises, determine whether the relation represents a function.
For the following exercises, determine if the relation represented in table form represents
 as a function of

For the following exercises, use the function
 represented in the table below.
For the following exercises, evaluate the function
 at the values
 and

For the following exercises, evaluate the expressions, given functions
 and

### Technology

For the following exercises, graph
  on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
 on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
 on the given domain. Determine the corresponding range. Show each graph.
For the following exercises, graph
 on the given domain. Determine the corresponding range. Show each graph.

###  Real-World Applications