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Inequalities are similar to equations in that they show a relationship between two expressions. We solve and graph inequalities in a similar way to equations. However, there are some differences that we will talk about in this chapter. The main difference is that for linear inequalities the answer is an interval of values whereas for a linear equation the answer is most often just one value. When writing inequalities we use the following symbols > is greater than $\ge$ is greater than or equal to < is less than $\le$ is less than or equal to

## Write and Graph Inequalities in One Variable on a Number Line

Let's start with the simple inequality x > 3 We read this inequality as "x is greater than 3." The solution is the set of all real numbers that are greater than three. We often represent the solution set of an inequality by a number line graph. Consider another simple inequality $x\le4$ We read this inequality as "x is less than or equal to 4." The solution is the set of all real numbers that equal four or less than four. We graph this solution set on the number line. In a graph, we use an empty circle for the endpoint of a strict inequality (x > 3) and a filled circle if the equal sign is included (x $latex \le$ 4).

### Example 1

Graph the following inequalities on the number line.
1. x < −3
2. $x\ge6$
3. x > 0
4. $x\le8$

### Solutions

1. The inequality x < −3 represents all real numbers that are less than −3. The number −3 is not included in the solution and that is represented by an open circle on the graph.
2. The inequality $x\ge6$ represents all real numbers that are greater than or equal to six. The number six is included in the solution and that is represented by a closed circle on the graph.
3. The inequality x > 0 represents all real numbers that are greater than zero. The number zero is not included in the solution and that is represented by an open circle on the graph.
4. The inequality $x\le8$ represents all real numbers that are less than or equal to eight. The number eight is included in the solution and that is represented by a closed circle on the graph.

### Example 2

Write the inequality that is represented by each graph.

### Solutions

1. x < −12
2. x > 540
3. x < 65
4. $x\ge83$
Inequalities appear everywhere in real life. Here are some simple examples of real-world applications.

### Example 3

Write each statement as an inequality and graph it on the number line.
1. You must maintain a balance of at least $2500 in your checking account to get free checking. 2. You must be at least 48 inches tall to ride the "Thunderbolt" Rollercoaster. 3. You must be younger than 3 years old to get free admission at the San Diego Zoo. 4. The speed limit on the interstate is 65 miles per hour. ### Solutions 1. The inequality is written as $x\ge2500$. The words "at least" imply that the value of$2500 is included in the solution set.
2. The inequality is written as $x\ge48$. The words "at least" imply that the value of 48 inches is included in the solution set.
3. The inequality is written as x < 3.
4. Speed limit means the highest allowable speed, so the inequality is written as $x\le65$.

# Solve an Inequality Using Addition

To solve an inequality we must isolate the variable on one side of the inequality sign. To isolate the variable, we use the same basic techniques used in solving equations. For inequalities of this type: xa < b or xa > b We isolate the x by adding the constant a to both sides of the inequality.

### Example 4

Solve each inequality and graph the solution set.
1. x−3 < 10
2. x−1 > −10
3. $x−1\le−5$
4. $x−20\ge14$

### Solutions

1. To solve the inequality x- 3 < 10 Simplify: x < 13
2. To solve the inequality x - 1 > -10
3. To solve the inequality  $x-1\le-5$.Simplify: $x\le-4$
4. To solve the inequality $x-20\le14$

# Solve an Inequality Using Subtraction

For inequalities of this type: x + 1 < b or x + 1 > b We isolate the x by subtracting the constant a on both sides of the inequality.

### Example 5

Solve each inequality and graph the solution set.
1. $x + 2 < 7$
2. $x+8\le−7$
3. $x+4>13$

### Solutions

1. To solve the inequality x + 2 < 7, subtract 2 on both sides of the inequality. x + 2 – 2 < 7 – 2 Simplify: x < 5
2. To solve the inequality $x+8\le–7$, subtract 8 on both sides of the inequality. $x + 8 – 8\le-7 – 8$ Simplify: [latex-display]x\ge–15[/latex-display]
3. To solve the inequality x + 4 > 13, subtract 4 on both sides of the inequality. x + 4 – 4 > 13 – 4 Simplify: x > 9

# Solve an Inequality Using Multiplication

Consider the problem: $latex \frac{x}{5} \le 3$ To find the solution we multiply both sides by 5: $latex (5) \frac{x}{5} \le 3 (5)$ We obtain $latex x\le 15$. The answer of an inequality can be expressed in four different ways:
1. Inequality notation The answer is simply expressed as x < 15.
2. Set notation The answer is x|x < 15. You read this as "the set of all values of x, such that x is a real number less than 15".
3. Interval notation uses brackets to indicate the range of values in the solution.The interval notation solution for our problem is (−∞,15). Interval notation also uses the concept of infinity ∞ and negative infinity −∞. Round or open brackets "(" and ")" indicate that the number next to the bracket is not included in the solution set. Square or closed brackets "[" and "]" indicate that the number next to the bracket is included in the solution set.
4. Solution graph shows the solution on the real number line. A closed circle on a number indicates that the number is included in the solution set. While an open circle indicates that the number is not included in the set. For our example, the solution graph is drawn here.

### Example 6

1. [−4,6] says that the solutions is all numbers between −4 and 6 including −4 and 6.
2. (8,24) says that the solution is all numbers between 8 and 24 but does not include the numbers 8 and 24.
3. [3,12) says that the solution is all numbers between 3 and 12, including 3 but not including 12.
4. (−5, ∞) says that the solution is all numbers greater that −5, not including −5.
5. (−∞,∞) says that the solution is all real numbers.

# Solving an Inequality Using Division

We solve the inequality $latex 2x \ge 12$by dividing both sides by 2: $latex \frac{2x}{2} \ge \frac{12}{2}$ Simplify: $latex x \ge 6$ Let's write the solution in the four different notations you just learned:
 Inequality notation $latex x \ge 6$ Set notation $latex {x | x \ge 6 }$ Interval notation $latex [6,$ ∞] Solution graph

# Multiplying and Dividing an Inequality by a Negative Number

We solve an inequality in a similar way to solving a regular equation. We can add or subtract numbers on both sides of the inequality. We can also multiply or divide positive numbers on both sides of an inequality without changing the solution. Something different happens if we multiply or divide by negative numbers. In this case, the inequality sign changes direction. For example, to solve −3x < 9 We divide both sides by –3. The inequality sign changes from < to > because we divide by a negative number. $latex \frac{-3x}{-3} \gt\ 6\frac{9}{-3}$ $latex x \gt\ -3$ We can explain why this happens with a simple example. We know that two is less than three, so we can write the inequality. 2 < 3 If we multiply both numbers by −1 we get −2 and −3, but we know that −2 is greater than −3. −2 > −3 You see that multiplying both sides of the inequality by a negative number caused the inequality sign to change direction. This also occurs if we divide by a negative number.

### Example 7

Solve each inequality. Give the solution in inequality notation and interval notation.
1. $latex 4x \lt 24$$\displaystyle{4}{x}lt{24}$
2. $latex -9x \le \frac{3}{5}$
3. $\displaystyle-{5}{x}\le{21}$
4. $\displaystyle{12}{x}\gt-{30}$

### Solutions

1. $\displaystyle{4}{x}\lt{24}$Divide both sides by 4:$\displaystyle\frac{{{4}{x}}}{{4}}\lt\frac{{24}}{{4}}$Simplify to get the answer: $\displaystyle{x}\lt{6}$
2. $\displaystyle-{9}{x}\le-\frac{{3}}{{5}}$Divide both sides by –9: $\displaystyle\frac{{-{9}{x}}}{{-{9}}}\ge-\frac{{3}}{{5}}(-\frac{{1}}{{9}})$. The direction of the inequality is changed.Simplify to get the answer: $\displaystyle{x}\ge\frac{{1}}{{15}}$
3. $\displaystyle-{5}{x}\le{21}$Divide both sides by –5: $\displaystyle\frac{{-{5}{x}}}{{-{{5}}}}\ge\frac{{21}}{{-{{5}}}}$Direction of the inequality is changed. Simplify to get the answer $\displaystyle{x}\ge-\frac{{21}}{{5}}$
4. $\displaystyle{12}{x}\gt-{30}$Divide both sides by 12:$\displaystyle\frac{{{12}{x}}}{{12}}\gt-\frac{{30}}{{12}}$Simplify to get the answer $\displaystyle{x}\gt-\frac{{5}}{{2}}$

### Example 8

Solve each inequality. Give the solution in inequality notation.
1. $\displaystyle\frac{{x}}{{2}}\gt{40}$
2. $\displaystyle\frac{{x}}{{-{3}}}\le-{12}$
3. $\displaystyle\frac{{x}}{{25}}\lt\frac{{3}}{{2}}$
4. $\displaystyle\frac{{x}}{{-{7}}}\ge{9}$

### Solutions

1. $\displaystyle\frac{{x}}{{2}}\gt{40}$Multiply both sides by 2:$\displaystyle{2}(\frac{{x}}{{2}})\gt{40}({2})$Simplify:$\displaystyle{x}\gt{80}$
2. $\displaystyle\frac{{x}}{{-{3}}}\le{12}$Multiply both sides by –3: $\displaystyle-{3}(\frac{{x}}{{-{3}}})\ge-{12}(-{3})$Direction of inequality is changed.Simplify:$\displaystyle{x}\ge{36}$
3. $\displaystyle\frac{{x}}{{25}}\lt\frac{{3}}{{2}}$Multiply both sides by 25$\displaystyle{25}(\frac{{x}}{{25}})\lt\frac{{3}}{{2}}({25})$Simplify: $\displaystyle{x}\lt\frac{{75}}{{2}}$ or 37.5
4. $\displaystyle\frac{{x}}{{-{7}}}\ge{9}$Multiply both sides by –7:$\displaystyle(-{7})\frac{{x}}{{-{7}}}\le{9}(-{7})$Direction of inequality is changed.Simplify: $\displaystyle{x}\le-{63}$

# Section Summary

• The answer to an inequality is often an interval of values. Common inequalities are:
• > is greater than
• ge is greater than or equal to
• > is less than
• le is less than or equal to
• Solving inequalities with addition and subtraction works just like solving an equation. To solve, we isolate the variable on one side of the equation.
• There are four ways to represent an inequality:
1. Equation notation x ge 2
2. Set notation x ge 2
3. Interval notation [2,∞) Closed brackets "[" and "]" mean inclusive, parentheses "("and ")" mean exclusive.
4. Solution graph
• When multiplying or dividing both sides of an inequality by a negative number, you need to reverse the inequality.

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