# Solving inequalities

When we add or subtract the same number on both sides of the truth of the inequality doesn't change.

This holds true for all numbers:

$\\ \begin{matrix} x> y \to \: \: \: \: \: \: \: \: \: \: \: \: \: \: & x< y \to\: \: \: \: \: \: \: \: \: \: \: \: \\ x+z> y+z\to\: \: &x+z< y+z\to \\ x-z> y-z \: \: \: \: \: \: \: \: & x-z< y-z\: \: \: \: \: \: \end{matrix}$

Example:

$\begin{array}{lcl} \: \: \: \: \, \, \, x+3> 9\\ x+3-3> 9-3\\ \; \; \; \; \; \; \; \: \: \, \, x> 6 \end{array}$

It is a little bit trickier when it comes to division and multiplication

When we multiply or divide an inequality by a positive integer, the truth of the inequality doesn't change.

$\\x> y \to \\ x\cdot z> y\cdot z\to \\\\ \frac{x}{z}> \frac{y}{z}\\ \\ If\; z> 0 \\$

When we multiply or divide an inequality by a negative integer, the sign of the inequality will be reversed (changed).

$\\x> y \to \\ x\cdot z< y\cdot z\to \\\\ \frac{x}{z}< \frac{y}{z}\\ \\ If\; z< 0 \\$

Example:

$\\ \frac{x}{-2}\geq 3\\\\ \frac{x}{-2}\cdot -2\geq 3\cdot -2\\\\ x\leq -6 \\$

Video lesson: Solve the inequality

$2-3x<14$