standard_description: Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
problem_activity: Activity
The purpose of this activity is for students to practice interpreting the solution of an inequality, especially those involving negative numbers. Students also practice representing inequalities written in natural language with symbols and number lines, and connecting the solution of an inequality with a number line. The work in this activity will be useful when students solve more complex inequalities in the associated Algebra 1 lesson.
Student Facing
Solve each inequality. Then, check your answer using a value that makes your solution true.
\(\text-2x<4\)
Solve the inequality.
Check your answer using a value that makes your solution true.
\(3x+5>6x-4\)
Solve the inequality.
Check your answer using a value that makes your solution true.
\(\text-3(x+1)\geq13\)
Solve the inequality.
Check your answer using a value that makes your solution true.
For each statement:
Use a number line to show which values satisfy the inequality.
Express the statement symbolically with an inequality.
The elevator can lift up to 1,200 pounds. Let
\(x\)
represent the weight being lifted by the elevator.
[IMG]
Over the course of the senator's term, her approval rating was always around 53% ranging 3% above or below that value. Let
\(x\)
represent the senator’s approval rating.
[IMG]
There's a minimum of 3 years of experience required. Let
\(x\)
represent the years of experience a candidate has.
[IMG]
Student Response
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Activity Synthesis
The goal of this discussion is to help students make sense of their solutions. Here are questions for discussion:
"After you checked your answers using a value from your solution, did any of your answers change?" (Yes. Initially I thought the solution was
\(x <\text-2\)
, but I used
\(\text-3\)
to check my answer and realized it was incorrect. When I substituted
\(\text-3\)
for
\(x\)
in the original inequality, I got
\(6<4\)
which is not true. This reminds me to change the inequality symbol when I multiply or divide by a negative number.)
"Could the elevator hold 1,300 pounds?" (No. It holds
up to
1,200, so this is the largest amount the elevator can hold.)
"Could the senator’s approval rating be 54%?" (Yes. The situation says the approval rating can be 3 points above and below 53%, and 54% fits that requirement.)
"How can the number line be useful in writing the solution to an inequality?" (It helps to see which values are included in the solution and which are not. It can also be used to check your answer after you solve an inequality.)
"On the graph of the minimum years of experience for a candidate, does it make sense to have an upper bound? What examples can you think of that would make this situation have an upper bound?" (Yes, because realistically a person cannot have infinity years of experience and that is what the current graph of the inequality reprsents. For example, many people retire after a certain age, so it would be hard for someone to have more than 50 years of experience. Also, the average lifespan is about 110 and people ususally start working as teens or young adults, so this creates another limit on how many years of experience someone can actually have.)


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