How to Solve Multi-Step Inequalities? (+FREE Worksheet!)

Related Topics

• How to Solve One-Step Equations
• How to Solve Multi-Step Equations
• How to Solve One-Step Inequalities
• How to Solve Systems of Equations
• How to Graph Single–Variable Inequalities

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.

Multi–Step Inequalities – Example 1:

Solve this inequality. \(4x-8 > 24\)

Solution:

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

Multi–Step Inequalities – Example 2:

Solve this inequality.  \(2x + 6 \leq10\)

Solution:

First subtract \(6\) from both sides: \(2x + 6 \leq10\) → \(2x+6−6≤10−6\)
Then simplify: \(2x+6−6≤10−6→2x≤4\)
Now divide both sides by \(2: \frac{2x}{2}≤\frac{4}{2} →x≤2\)

Multi–Step Inequalities – Example 3:

Solve this inequality. \(2x-2≤6\)

Solution:

First add \(2\) to both sides: \(2x-2≤6\) → \(2x-2+2≤6+2\)

Then simplify: \(2x-2+2≤6+2→2x≤8\)
Now, divide both sides by \(2: \frac{2x}{2}≤\frac{8}{2}→x≤4\)

Multi–Step Inequalities – Example 4:

Solve this inequality. \(-2x-4 < 8\)

Solution:

First add \(4\) to both sides: \(-2x-4 < 8\) → \(-2x-4+4 < 8+4\)
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 for Solving Multi–Step Inequalities

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. \(\color{blue}{\frac{2x + 6}{4} ≤ 10} \\ \)

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Answers

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. \(\color{blue}{x ≤ 17}\)