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6.1: Solving Inequalities

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Learning Objectives

  • Write and graph inequalities in one variable on a number line.
  • Solve inequalities using addition and subtraction.
  • Solve inequalities using multiplication and division.
  • Solve multi-step inequalities.

Introduction

Dita has a budget of $350 to spend on a rental car for an upcoming trip, but she wants to spend as little of that money as possible. If the trip will last five days, what range of daily rental rates should she be willing to consider?

Like equations, inequalities show a relationship between two expressions. We solve and graph inequalities in a similar way to equations—but when we solve an inequality, the answer is usually a set of values instead of 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 with a number line graph.

Consider another simple inequality: x \le 4.

We read this inequality as “x is less than or equal to 4.” The solution is the set of all real numbers that are equal to four or less than four. We can graph this solution set on the number line.

Notice that we use an empty circle for the endpoint of a strict inequality (like x > 3), and a filled circle for one where the equals sign is included (like x \le 4).

Example 1

Graph the following inequalities on the number line.

a) x< -3

b) x \ge 6

c) x > 0

d) x \le 8

Solution

a) The inequality x < -3 represents all numbers that are less than -3. The number -3 is not included in the solution, so it is represented by an open circle on the graph.

b) The inequality x \ge 6 represents all numbers that are greater than or equal to 6. The number 6 is included in the solution, so it is represented by a closed circle on the graph.

c) The inequality x > 0 represents all numbers that are greater than 0. The number 0 is not included in the solution, so it is represented by an open circle on the graph.

d) The inequality x \le 8 represents all numbers that are less than or equal to 8. The number 8 is included in the solution, so it is represented by a closed circle on the graph.

Example 2

Write the inequality that is represented by each graph.

a)

b)

c)

d)

Solution

a) x \le -12

b) x >540

c) x < 6.5

d) x \ge 85

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.

a) You must maintain a balance of at least $2500 in your checking account to get free checking.

b) You must be at least 48 inches tall to ride the “Thunderbolt” Rollercoaster.

c) You must be younger than 3 years old to get free admission at the San Diego Zoo.

d) The speed limit on the interstate is 65 miles per hour or less.

Solution

a) The words “at least” imply that the value of $2500 is included in the solution set, so the inequality is written as x \ge 2500.

b) The words “at least” imply that the value of 48 inches is included in the solution set, so the inequality is written as x \ge 48.

c) The inequality is written as x < 3.

d) Speed limit means the highest allowable speed, so the inequality is written as x \le 65.

Solving Inequalities Using Addition and Subtraction

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.

We can solve some inequalities by adding or subtracting a constant from one side of the inequality.

Example 4

Solve each inequality and graph the solution set.

a) x-3<10

b) x-20 \ge 14

c) x+8 \le -7

d) x+4 > 13

Solution

a) Starting inequality: x-3 < 10

Add 3 to both sides of the inequality: x - 3 + 3 < 10 + 3

Simplify: x < 13

b) Starting inequality: x - 20 \le 14

Add 20 to both sides of the inequality: x - 20 + 20 \le 14 + 20

Simplify: x \le 34

c) Starting inequality: x+8 \le -7

Subtract 8 from both sides of the inequality: x + 8 - 8 \le -7 - 8

Simplify: x\le -15

d) Starting inequality: x+4 > 13

Subtract 4 from both sides of the inequality: x + 4 - 4 > 13 - 4

Simplify: x > 9

Solving Inequalities Using Multiplication and Division

We can also solve inequalities by multiplying or dividing both sides by a constant. For example, to solve the inequality 5x<3, we would divide both sides by 5 to get x < \frac{3}{5}.

However, something different happens when we multiply or divide by a negative number. We know, for example, that 5 is greater than 3. But if we multiply both sides of the inequality 5>3 by -2, we get -10 > -6. And we know that’s not true; -10 is less than -6.

This happens whenever we multiply or divide an inequality by a negative number, and so we have to flip the sign around to make the inequality true. For example, to multiply 2 < 4 by -3, first we multiply the 2 and the 4 each by -3, and then we change the < sign to a > sign, so we end up with -6 > -12.

The same principle applies when the inequality contains variables.

Example 5

Solve each inequality.

a) 4x < 24

b) -5x \le 21

c) \frac{x}{25} < \frac{3}{2}

d) \frac{x}{-7} \ge 9

Solution

a) Original problem: 4x < 24

Divide both sides by 4: \frac{4x}{4} < \frac{24}{4}

Simplify: x < 6

b) Original problem: -5x \le 21

Divide both sides by -5 : \frac{-5x}{-5} \ge \frac{21}{-5} Flip the inequality sign.

Simplify: x \ge -\frac{21}{5}

c) Original problem: \frac{x}{25} < \frac{3}{2}

Multiply both sides by 25: 25 \cdot \frac{x}{25} < \frac{3}{2} \cdot 25

Simplify: x < \frac{75}{2} or x < 37.5

d) Original problem: \frac{x}{-7} \ge 9

Multiply both sides by -7: -7 \cdot \frac{x}{-7} \le 9 \cdot (-7) Flip the inequality sign.

Simplify: x \le -63

Solving Multi-Step Inequalities

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 6

Solve each of the following inequalities and graph the solution set.

a) \frac{9x}{5}-7 \ge -3x + 12

b) -25x + 12 \le -10x - 12

Solution

a) Original problem: \frac{9x}{5}-7 \ge -3x+12

Add 3x to both sides: \frac{9x}{5} + 3x - 7 \ge -3x+3x+12

Simplify: \frac{24x}{5}-7 \ge 12

Add 7 to both sides: \frac{24x}{5}-7+7 \ge 12+7

Simplify: \frac{24x}{5} \ge 19

Multiply 5 to both sides: 5 \cdot \frac{24x}{5} \ge 5 \cdot 19

Simplify: 24x \ge 95

Divide both sides by 24: \frac{24x}{24} \ge \frac{95}{24}

Simplify: x \ge \frac{95}{24} Answer

Graph:

b) Original problem: -25x+12 \le -10x-12

Add 10x to both sides: -25x+10x+12 \le -10x+10x-12

Simplify: -15x+12 \le -12

Subtract 12: -15x+12-12\le -12-12

Simplify: -15x \le -24

Divide both sides by -15: \frac{-15x}{-15} \ge \frac{-24}{-15} flip the inequality sign

Simplify: x \ge \frac{8}{5} Answer

Graph:

Example 7

Solve the following inequalities.

a) 4x-2(3x-9) \le -4(2x-9)

b) \frac{5x-1}{4} > -2(x+5)

Solution

a) Original problem: 4x-2(3x-9) \le -4(2x-9)

Simplify parentheses: 4x-6x+18 \le -8x+36

Collect like terms: -2x+18 \le -8x+36

Add 8x to both sides: -2x+8x+18 \le -8x+8x+36

Simplify: 6x+18 \le 36

Subtract 18: 6x+18-18 \le 36-18

Simplify: 6x \le 18

Divide both sides by 6: \frac{6x}{6} \le \frac{18}{6}

Simplify: x \le 3 Answer

b) Original problem: \frac{5x-1}{4} > -2(x+5)

Simplify parenthesis: \frac{5x-1}{4} > -2x-10

Multiply both sides by 4: 4 \cdot \frac{5x-1}{4} > 4 (-2x-10)

Simplify: 5x-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: 13x > -39

Divide both sides by 13: \frac{13x}{13} > -\frac{39}{13}

Simplify: x>-3 Answer

Further Practice

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.

Lesson Summary

  • 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.

Review Questions

  1. Write the inequality represented by the graph.
  2. Write the inequality represented by the graph.
  3. Write the inequality represented by the graph.
  4. Write the inequality represented by the graph.

Graph each inequality on the number line.

  1. x < -35
  2. x > -17
  3. x \ge 20
  4. x \le 3

Solve each inequality and graph the solution on the number line.

  1. x-5 < 35
  2. x+15 \ge -60
  3. x-2 \le 1
  4. x-8 > -20
  5. x+11>13
  6. x+65<100
  7. x-32 \le 0
  8. x+68 \ge 75

Solve each inequality. Write the solution as an inequality and graph it.

  1. 3x \le 6
  2. \frac{x}{5} > -\frac{3}{10}
  3. -10x>250
  4. \frac{x}{-7} \ge -5

Solve each multi-step inequality.

  1. x-5 > 2x+3
  2. 2(x-3) \le 3x-2
  3. \frac{x}{3} < x+7
  4. \frac{3(x-4)}{12} \le \frac{2x}{3}
  5. 2 \left ( \frac{x}{4} + 3\right ) > 6(x-1)
  6. 9x+4 \le -2 \left ( x+\frac{1}{2} \right )

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