Inequality Expression

Question

what happens when an inequality expression is divided by the same number. Did the sign stay the same each time or change directions? If the sign changed, explain what made the direction of the sign change. Describe a rule for this.

Answer 1

When you multiply or divide by a positive number, the inequality is preserved.

When you multiply or divide by a negative number, the inequality is reversed.

Answer 2

Say you have real numbers $x$ and $y$ with $x \geq y$, and a function $f: \mathbb{R} \rightarrow \mathbb{R}$. If the function $f$ is increasing then (by definition) we have $f(x) \geq f(y)$. Similarly if $f$ is a decreasing function then we have $f(x) \leq f(y)$. The map $t \mapsto \lambda t$ (for $\lambda$ a fixed real number) is increasing if $\lambda \geq 0$ and decreasing if $\lambda \leq 0$. So mulitplication by a non-negative number (which includes division by a positive number) preserves $\geq$ whilst multiplication by a non-positive number (for example, dividing by a negative) reverses $\geq$.