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6.3: Multi-Step Inequalities

Difficulty Level: At Grade Created by: CK-12
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What if you had an inequality with an unknown variable on both sides like \begin{align*}2(x - 2) > 3x - 5\end{align*}2(x2)>3x5? How could you isolate the variable to find its value? After completing this Concept, you'll be able to solve multi-step inequalities like this one.

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CK-12 Foundation: 0603S Solving Multi-Step Inequalities (H264)

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For additional practice solving inequalities, try the online game at http://www.aaamath.com/equ725x7.htm#section2. If you’re having a hard time with multi-step inequalities, the video at http://www.schooltube.com/video/aa66df49e0af4f85a5e9/MultiStep-Inequalities will walk you through a few.


In the last two sections, we considered very simple inequalities which required one step to obtain the solution. However, most inequalities require several steps to arrive at the solution. As with solving equations, we must use the order of operations to find the correct solution. In addition, remember that when we multiply or divide the inequality by a negative number, the direction of the inequality changes.

The general procedure for solving multi-step inequalities is almost exactly like the procedure for solving multi-step equations:

  1. Clear parentheses on both sides of the inequality and collect like terms.
  2. Add or subtract terms so the variable is on one side and the constant is on the other side of the inequality sign.
  3. Multiply and divide by whatever constants are attached to the variable. Remember to change the direction of the inequality if you multiply or divide by a negative number.

Example A

Solve the inequality \begin{align*}\frac{9x}{5}-7 \ge -3x + 12\end{align*}9x573x+12 and graph the solution set.


Original problem: \begin{align*}\frac{9x}{5}-7 \ge -3x+12\end{align*}9x573x+12

Add \begin{align*}3x\end{align*}3x to both sides: \begin{align*}\frac{9x}{5} + 3x - 7 \ge -3x+3x+12\end{align*}9x5+3x73x+3x+12

Simplify: \begin{align*}\frac{24x}{5}-7 \ge 12\end{align*}24x5712

Add 7 to both sides: \begin{align*}\frac{24x}{5}-7+7 \ge 12+7\end{align*}24x57+712+7

Simplify: \begin{align*}\frac{24x}{5} \ge 19\end{align*}24x519

Multiply 5 to both sides: \begin{align*}5 \cdot \frac{24x}{5} \ge 5 \cdot 19\end{align*}524x5519

Simplify: \begin{align*}24x \ge 95\end{align*}24x95

Divide both sides by 24: \begin{align*}\frac{24x}{24} \ge \frac{95}{24}\end{align*}24x249524

Simplify: \begin{align*}x \ge \frac{95}{24}\end{align*}x9524 Answer


Example B

Solve the inequality \begin{align*}-25x + 12 \le -10x - 12\end{align*}25x+1210x12 and graph the solution set.


Original problem: \begin{align*}-25x+12 \le -10x-12\end{align*}25x+1210x12

Add \begin{align*}10x\end{align*}10x to both sides: \begin{align*}-25x+10x+12 \le -10x+10x-12\end{align*}25x+10x+1210x+10x12

Simplify: \begin{align*}-15x+12 \le -12\end{align*}15x+1212

Subtract 12: \begin{align*}-15x+12-12\le -12-12\end{align*}15x+12121212

Simplify: \begin{align*}-15x \le -24\end{align*}15x24

Divide both sides by -15: \begin{align*}\frac{-15x}{-15} \ge \frac{-24}{-15}\end{align*}15x152415 flip the inequality sign

Simplify: \begin{align*}x \ge \frac{8}{5}\end{align*}x85 Answer


Example C

Solve the inequality \begin{align*}4x-2(3x-9) \le -4(2x-9)\end{align*}4x2(3x9)4(2x9).


Original problem: \begin{align*}4x-2(3x-9) \le -4(2x-9)\end{align*}4x2(3x9)4(2x9)

Simplify parentheses: \begin{align*}4x-6x+18 \le -8x+36\end{align*}4x6x+188x+36

Collect like terms: \begin{align*}-2x+18 \le -8x+36\end{align*}2x+188x+36

Add \begin{align*}8x\end{align*}8x to both sides: \begin{align*}-2x+8x+18 \le -8x+8x+36\end{align*}2x+8x+188x+8x+36

Simplify: \begin{align*}6x+18 \le 36\end{align*}6x+1836

Subtract 18: \begin{align*}6x+18-18 \le 36-18\end{align*}6x+18183618

Simplify: \begin{align*}6x \le 18\end{align*}6x18

Divide both sides by 6: \begin{align*}\frac{6x}{6} \le \frac{18}{6}\end{align*}6x6186

Simplify: \begin{align*}x \le 3\end{align*}x3 Answer

Watch this video for help with the Examples above.

CK-12 Foundation: Solving Multi-Step Inequalities


  • The answer to an inequality is usually an interval of values.
  • Solving inequalities works just like solving an equation. To solve, we isolate the variable on one side of the equation.
  • When multiplying or dividing both sides of an inequality by a negative number, you need to reverse the inequality.

Guided Practice

Solve the inequality \begin{align*}\frac{5x-1}{4} > -2(x+5)\end{align*}5x14>2(x+5).


Original problem: \begin{align*}\frac{5x-1}{4} > -2(x+5)\end{align*}5x14>2(x+5)

Simplify parenthesis: \begin{align*}\frac{5x-1}{4} > -2x-10\end{align*}5x14>2x10

Multiply both sides by 4: \begin{align*}4 \cdot \frac{5x-1}{4} > 4 (-2x-10)\end{align*}45x14>4(2x10)

Simplify: \begin{align*}5x-1 > -8x-40\end{align*}5x1>8x40

Add \begin{align*}8x\end{align*}8x to both sides: \begin{align*}5x + 8x - 1 >- 8x + 8x - 40\end{align*}5x+8x1>8x+8x40

Simplify: \begin{align*}13x-1>-40\end{align*}13x1>40

Add 1 to both sides: \begin{align*}13x-1+1>-40+1\end{align*}13x1+1>40+1

Simplify: \begin{align*}13x > -39\end{align*}13x>39

Divide both sides by 13: \begin{align*}\frac{13x}{13} > -\frac{39}{13}\end{align*}13x13>3913

Simplify: \begin{align*}x>-3\end{align*}x>3 Answer


Solve each multi-step inequality.

  1. \begin{align*}3x-5<x+3\end{align*}3x5<x+3
  2. \begin{align*}x-5 > 2x+3\end{align*}x5>2x+3
  3. \begin{align*}2(x-3) \le 3x-2\end{align*}2(x3)3x2
  4. \begin{align*}3(x+1) \ge 2x+5\end{align*}3(x+1)2x+5
  5. \begin{align*}2(x-9) \ge -1(4x+7)\end{align*}
  6. \begin{align*}\frac{x}{3} < x+7\end{align*}
  7. \begin{align*}\frac{x}{4} < 2x-21\end{align*}
  8. \begin{align*}\frac{3(x-4)}{12} \le \frac{2x}{3}\end{align*}
  9. \begin{align*}2 \left ( \frac{x}{4} + 3\right ) > 6(x-1)\end{align*}
  10. \begin{align*}9x+4 \le -2 \left ( x+\frac{1}{2} \right )\end{align*}

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linear equation

A linear equation is an equation between two variables that produces a straight line when graphed.

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Date Created:
Aug 13, 2012
Last Modified:
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