The inequality symbol flips because you have to keep the inequality as a true comparison. For example, if you have [katex] 3 < 5 [/katex] and you multiply both sides of the inequality by [katex] -1 [/katex] you will get [katex] -3 < -5. [/katex] This is no longer a true inequality comparison. In order to make it true, you will have to flip the inequality symbol, so [katex] < [/katex] becomes [katex] >. [/katex] [katex] -3 < -5 [/katex] becomes [katex] -3 > -5. [/katex]