# Multi–Step Inequalities To solve multi-step inequalities you need to do more than one math operation. Learn how to solve Multi-Step inequalities in few simple steps.

## Step by step guide to solve multi-step inequalities

• Isolate the variable similar to equations.
• Simplify using the inverse of addition or subtraction.
• Simplify further by using the inverse of multiplication or division.
• For dividing or multiplying both sides by negative numbers, flip the direction of the inequality sign.

### Example 1:

Solve this inequality. $$4x-8 > 24$$

Solution:

First add $$8$$ to both sides: $$4x-8+8 > 24+8$$
Then simplify: $$4x-8+8 > 24+8→4x > 32$$
Now divide both sides by $$4: \frac{4x}{4} > \frac{32}{4 } →x > 8$$

### Example 2:

Solve this inequality.  $$2x + 6 \leq10$$

Solution:

First subtract $$6$$ from both sides: $$2x+6−6≤10−6$$
Then simplify: $$2x+6−6≤10−6→2x≤4$$
Now divide both sides by $$2: \frac{2x}{2}≤\frac{4}{2} →x≤2$$

### Example 3:

Solve this inequality. $$2x-2≤6$$

Solution:

First add $$2$$ to both sides: $$2x-2+2≤6+2→2x≤8$$
Now, divide both sides by $$2: 2x≤8→x≤4$$

### Example 4:

Solve this inequality. $$-2x-4 < 8$$

Solution:

First add $$4$$ to both sides: $$-2x-4+4 < 8+4$$
Then simplify: $$-2x-4+4 < 8+4→-2x < 12$$
Now divide both sides by $$-2$$ (Remember, for dividing or multiplying both sides by negative numbers, flip the direction of the inequality sign.)

$$\frac{-2x}{-2} > \frac{12}{-2 } →x > -6$$

## Exercises

### Solve each inequality.

1. $$\color{blue}{\frac{9x}{7} – 7 < 2} \\$$
2. $$\color{blue}{\frac{4x + 8}{2} ≤ 12} \\$$
3. $$\color{blue}{\frac{3x – 8}{7} > 1} \\$$
4. $$\color{blue}{–3 (x – 7) > 21} \\$$
5. $$\color{blue}{4 + \frac{x}{3} < 7} \\$$
6. $$\color{blue}{\frac{2x + 6}{4} ≤ 10} \\$$

1. $$\color{blue}{x < 7 }$$
2. $$\color{blue}{x ≤ 4 }$$
3. $$\color{blue}{x > 5}$$
4. $$\color{blue}{x < 0 }$$
5. $$\color{blue}{x < 9}$$
6. $$\color{blue}{x ≤ 17}$$ 