Copyright © Cengage Learning. All rights reserved.

2Equations and Inequalities

Copyright © Cengage Learning. All rights reserved.

Section 2.72.7

Solving Linear Inequalities in One Variable

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Objectives

1. Solve a linear inequality in one variable using the properties of inequality and graph the solution on a number line.

2. Solve a compound linear inequality in one variable.

3. Solve an application involving a linear inequality in one variable.

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Solve a linear inequality in one variable using the properties of inequality and graph the solution on a number line

1.

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Solving an Inequality

Recall the meaning of the following symbols.

Inequality Symbols means “is less than” means “is greater than” means “is less than or equal to” means “is greater than or equal to”

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Solving an Inequality

A solution of an inequality is any number that makes the inequality true.

The number 2 is a solution of the inequality x 3because 2 3.

This inequality has many more solutions, because any real number that is less than or equal to 3 will satisfy it.

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Solving an Inequality

Using a graph to solve inequality: x 3

The red arrow in Figure 2-16 indicates all those points with coordinates that satisfy the inequality x 3.

The bracket at the point with coordinate 3 indicates that the number 3 is a solution of the inequality x 3.

Figure 2-16

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Solving an Inequality

The graph of the inequality x 1 appears in Figure 2-17.

The red arrow indicates all those points whose coordinates satisfy the inequality.

The parenthesis at the point with coordinate 1 indicates that 1 is not a solution of the inequality x 1.

Figure 2-17

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Solving an Inequality

Addition Property and Subtraction Property of InequalitySuppose a, b, and c are real numbers.

•If a b, then a + c b + c.•If a b, then a – c b – c.

Similar statements can be made for the symbols , , and .

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Solving an Inequality

The addition property of inequality can be stated this way: If any quantity is added to both sides of an inequality, the resulting inequality has the same direction as the original Inequality.

The subtraction property of inequality can be stated this way: If any quantity is subtracted from both sides of an inequality, the resulting inequality has the same direction as the original inequality.

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Example

Solve 2x + 5 x – 4 and graph the solution on a number line.

Solution:To isolate the x on the left side of the sign, we proceed as if we were solving an equation.

2x + 5 x – 42x + 5 – 5 x – 4 – 5 2x x – 9 2x – x x – 9 – x x –9

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Example – Solution

The graph of the solution (see Figure 2-18) includes all points to the right of –9 but does not include –9 itself. For this reason, we use a parenthesis at –9.

Figure 2-18

cont’d

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Solving an Inequality

If both sides of the true inequality 6 9 are multiplied or divided by a positive number, such as 3, another true inequality results.

6 9 6 9

3 6 3 9

18 27 2 3

The inequalities 18 27 and 2 3 are true.

Multiply both sides by 3. Divide both sides by 3.

True. True.

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Solving an Inequality

However, if both sides of 6 9 are multiplied or divided by a negative number, such as –3, the direction of the inequality symbol must be reversed to produce another true inequality.

6 9 6 9

–3 6 –3 9

–18 –27 –2 –3

Multiply both sidesby –3 and reversethe direction of theinequality.

Divide both sides by –3 and reverse the direction of the inequality.

True. True.

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Solving an Inequality

The inequality –18 –27 is true, because –18 lies to the right of –27 on the number line.

The inequality –2 –3 is true, because –2 lies to the right of –3 on the number line.

This example suggests the multiplication and division properties of inequality.

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Solving an Inequality

Multiplication Property of InequalitySuppose a, b, and c are real numbers.

If a b and c 0, then ac bc.If a b and c 0, then ac bc.

Division Property of InequalitySuppose a, b, and c are real numbers.

If a b and c 0, then .

If a b and c 0, then .

Similar statements can be made for the symbols , , and .

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Solving an Inequality

The multiplication property of inequality can be stated this way:

If unequal quantities are multiplied by the same positive quantity, the results will be unequal and in the same direction as the original inequality.

If unequal quantities are multiplied by the same negative quantity, the results will be unequal but in the opposite direction of the original inequality.

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Solving an Inequality

The division property of inequality can be stated this way:

If unequal quantities are divided by the same positive quantity, the results will be unequal and in the same direction as the original inequality.

If unequal quantities are divided by the same negative quantity, the results will be unequal but in the opposite direction of the original inequality.

To divide both sides of an inequality by a nonzero number c, we could instead multiply both sides by .

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Solving an Inequality

Note that the procedures for solving inequalities are the same as for solving equations, except that we must reverse the inequality symbol whenever we multiply or divide by a negative number.

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Solve a compound linear inequality in one variable2.

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Solving a Compound Inequality

Two inequalities often can be combined into a double inequality or compound inequality to indicate that numbers lie between two fixed values.

For example, the inequality 2 x 5 indicates that x is greater than 2 and that x is also less than 5. The solution of

2 x 5 consists of all numbers that lie between 2 and 5.

The graph of this set (called an interval) appears in Figure 2-21.

Figure 2-21

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Example

Solve –4 2(x – 1) 4, and graph the solution on the number line.

Solution:To isolate x in the center, we proceed as if we were solving an equation with three parts: a left side, a center, and a right side.

–4 2(x – 1) 4

–4 2x – 2 4 Use the distributive property to remove parentheses.

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Example – Solution–2 2x 6

–1 x 3

The graph of the solution appears in Figure 2-22.

Add 2 to all three parts.

Divide all three parts by 2.

Figure 2-22

cont’d

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Solve an application involving a linear inequality in one variable3.

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Solving an Inequality

When solving applications, there are certain words that help us translate a sentence into a mathematical inequality.

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A student has scores of 72, 74, and 78 points on three mathematics examinations. How many points does he need on his last exam to earn a B or better, an average of at least 80 points?

Solution:We can let x represent the score on the fourth (last) exam. To find the average grade, we add the four scores and divide by 4.

To earn a B, this average must be greater than or equal to 80 points.

Example – Grades

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We can solve this inequality for x.

224 + x 320

x 96

To earn a B, the student must score at least 96 points.

Add.

Multiply both sides by 4.

Subtract 224 from both sides.

cont’dExample – Solution

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Your Turn

Solve the following inequalities for x and graph the solution:

1. 2x + 5 < 13

2. -4 < 3x + 5 < 13