Multi–Step Inequalities

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

Download Multi-Step Inequalities Worksheet

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. \(\color{blue}{x ≤ 17}\)

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