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

$$2-3x<14$$