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# Multi-Step Inequalities

## Solve inequalities with fractions and distribution

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Multi-Step Inequalities

What if you had an inequality with an unknown variable on both sides like 2(x2)>3x5\begin{align*}2(x - 2) > 3x - 5\end{align*}? 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.

### Try This

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.

### Guidance

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

Solution

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

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

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

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

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

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

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

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

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

Graph:

#### Example B

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

Solution:

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

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

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

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

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

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

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

Graph:

#### Example C

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

Solution:

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

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

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

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

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

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

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

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

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

Watch this video for help with the Examples above.

### Vocabulary

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

Solution:

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

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

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

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

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

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

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

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

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

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

### Practice

Solve each multi-step inequality.

1. 3x5<x+3\begin{align*}3x-5
2. x5>2x+3\begin{align*}x-5 > 2x+3\end{align*}
3. 2(x3)3x2\begin{align*}2(x-3) \le 3x-2\end{align*}
4. 3(x+1)2x+5\begin{align*}3(x+1) \ge 2x+5\end{align*}
5. 2(x9)1(4x+7)\begin{align*}2(x-9) \ge -1(4x+7)\end{align*}
6. x3<x+7\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*}

### Vocabulary Language: English

linear equation

linear equation

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