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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 as follows.
1. Clear parenthesis 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.

## Solve a Two-Step Inequality

### Example 1

Solve each of the following inequalities and graph the solution set.
1. 6x − 5 < 10
2. −9x < −5x − 15
3. $\displaystyle-\frac{{{9}{x}}}{{5}}$$latex \le$ 24

#### Solution

1. 6x – 5 < 10Add 5 to both sides: 6 x –5 + 5 < 10 +5Simplify: 6 x < 15Divide both sides by 6: [latex-display]\displaystyle\frac{{{6}{x}}}{{6}}\lt\frac{{15}}{{6}}[/latex-display] Simplify: $\displaystyle{x}\lt\frac{{5}}{{2}}$
2. –9x < –5x – 15 Add 5x to both sides: –9x + 5x < –5x + 5x – 15 Simplify: –4x < –15
Divide both sides by –4: [latex-display]\displaystyle\frac{{-{4}{x}}}{{-{{4}}}}\gt-\frac{{15}}{{-{{4}}}}[/latex-display] Inequality sign is flipped Simplify: [latex-display]\displaystyle{x}\gt\frac{{15}}{{4}}[/latex-display] 3. $\displaystyle-\frac{{{9}{x}}}{{5}}$$latex \le$ 24 Multiply both sides by 5: [latex-display]\displaystyle{5}\frac{{-{9}{x}}}{{5}}\le{24}({5})[/latex-display] Simplify: –9 x $latex \le$120 Divide both sides by –9: [latex-display]\displaystyle\frac{{-{9}{x}}}{{-{{9}}}}\ge\frac{{120}}{{-{{9}}}}[/latex-display] Inequality sign is flipped. Simplify: [latex-display]\displaystyle{x}\ge-\frac{{40}}{{3}}[/latex-display]

# Solve a Multi-Step Inequality

### Example 2

Each of the following inequalities and graph the solution set.
1. $\displaystyle\frac{{{9}{x}}}{{5}}-{7}\ge-{3}{x}+{12}$
2. −25x+12$latex \le$ −10x−12

### Solutions

[latex-display]\displaystyle\frac{{{9}{x}}}{{5}}-{7}\ge-{3}{x}+{12}[/latex-display]
1. Add 3x to both sides: $\displaystyle\frac{{{9}{x}}}{{5}}+{3}{x}-{7}\ge-{3}{x}+{3}{x}+{12}$Simplify: $\displaystyle\frac{{{24}{x}}}{{5}}-{7}\ge{12}$Add 7 to both sides: [latex-display]\displaystyle\frac{{{24}{x}}}{{5}}-{7}+{7}\ge{12}+{7}[/latex-display] Simplify: [latex-display]\displaystyle\frac{{{24}{x}}}{{5}}\ge{19}[/latex-display] Multiply both sides by 5: [latex-display]\displaystyle{5}\frac{{{24}{x}}}{{5}}\ge{19}({5})[/latex-display] Simplify: [latex-display]\displaystyle{24}{x}\ge{95}[/latex-display] Divide both sides by 24: [latex-display]\displaystyle\frac{{{24}{x}}}{{24}}\ge\frac{{95}}{{24}}[/latex-display] Simplify:
2. –25x + 12$latex \le$ –10x – 12Add 10x to both sides: –25x + 10x + 12$latex \le$ –10x + 10x – 12Simplify: –15x + 12 $latex \le$ –12Subtract 12 from both sides: –15x + 12 – 12$latex \le$ –12 – 12 Simplify: –15x $latex \le$ –24 Divide both sides by –15: $\displaystyle\frac{{-{15}{x}}}{{-{{15}}}}\ge\frac{{-{24}}}{{-{{15}}}}$ Inequality sign is flipped. Simplify: $\displaystyle{x}\ge\frac{{8}}{{5}}$

### Example 3

Solve the following inequalities.
1. 4x − 2(3x − 9)$latex \le$ −4(2x − 9)
2. $\displaystyle\frac{{{5}{x}-{1}}}{{4}}$> −2(x + 5)

### Solutions

1. 4x − 2(3x − 9)$latex \le$ −4(2x − 9)
Simplify parentheses: 4 – 6 x + 18$latex \le$ –8x + 36 Collect like terms: –2 x + 18$latex \le$ –8x + 36 Add 8x to both sides: –2x + 8x + 18$latex \le$–8x + 8x + 36 Simplify: 6 x + 18$latex \le$ 36 Subtract 18 from both sides: 6 x + 18 –18$latex \le$ 36 –18 Simplify: 6 x $latex \le$ 18 Divide both sides by 6: [latex-display]\displaystyle\frac{{{6}{x}}}{{6}}\le\frac{{18}}{{6}}[/latex-display] Simplify: x $latex \le$ 3
1. $\displaystyle\frac{{{5}{x}-{1}}}{{4}}\gt-{2}{({x}+{5})}$Simplify parenthesis: $\displaystyle\frac{{{5}{x}-{1}}}{{4}}\gt-{2}{x}-{10}$Multiply both sides by 4: $\displaystyle{4}\frac{{{5}{x}-{1}}}{{4}}\gt{4}{(-{2}{x}-{10})}$Simplify: 5 x –1 > –8x –40 Add 8x to both sides: 5x +8x –1 > 8x + 8x – 40 Simplify: 13x –1 > – 40 Add 1 to both sides: 13x –1 + 1 > – 40 + 1 Simplify: 13 x > – 39 Divide both sides by 13: $\displaystyle\frac{{{13}{x}}}{{13}}\gt-\frac{{39}}{{13}}$ Simplify: x > –3

# Identify the Number of Solutions of an Inequality

Inequalities can have:
• A set that has an infinite number of solutions.
• No solutions
• A set that has a discrete number of solutions.

## Infinite Number of Solutions

The inequalities we have solved so far all have an infinite number of solutions. In the last example, we saw that the inequality $\displaystyle\frac{{{5}{x}-{1}}}{{4}}\gt-{2}{({x}+{5})}$has the solution x > −3 This solution says that all real numbers greater than −3 make this inequality true. You can see that the solution to this problem is an infinite set of numbers.

## No solutions

Consider the inequality x − 5 > x + 6 This simplifies to −5 > 6 This statements is not true for any value of x. We say that this inequality has no solution.

## Discrete solutions

So far we have assumed that the variables in our inequalities are real numbers. However, in many real life situations we are trying to solve for variables that represent integer quantities, such as number of people, number of cars or number of ties.

### Example 4

Raul is buying ties and he wants to spend $200 or less on his purchase. The ties he likes the best cost$50. How many ties could he purchase?

### Example 7

The width of a rectangle is 20 inches. What must the length be if the perimeter is at least 180 inches?

### Solution

#### Step 1

Width = 20 inches; perimeter is at least 180 inches. What is the smallest length that gives that perimeter? Let x = length of the rectangle

#### Step 2

Formula for perimeter is Perimeter = 2 × length + 2 × width Since the perimeter must be at least 180 inches, we have the following equation. 2 x +2(20)$latex \ge$ 180

### Step 3

We solve the inequality. Simplify. 2 x + 40$latex \ge$ 180 Subtract 40 from both sides. 2 x $latex \ge$ 140 Divide both sides by 2. x $latex \ge$e 70 Answer: The length must be at least 70 inches.

#### Step 4

If the length is at least 70 inches and the width is 20 inches, then the perimeter can be found by using this equation. 2(70) + 2(20) = 180 inches The answer checks out.

# Section Summary

• The general procedure for solving multi-step inequalities is as follows.
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.
• Inequalities can have multiple solutions, no solutions, or discrete solutions.

ck12, Algebra, Linear Inequalities, " Solving One-Step Inequalities," licensed under a CC BY-NC 3.0 license.