# 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:

$x> y \to x< y \to$

$x+z> y+z\to x+z< y+z\to$

$x-z> y-z \,\,\, x-z< y-z$

Example

$x+3> 9$

$x+3-3> 9-3$

$x> 6$

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

$2-3x<14$