1 Note that the “>” can be replaced by ,

1

Note that the “>” can be replaced by , <, or .

Examples: Linear inequalities in one variable.

2x – 2 < 6x – 5

A linear inequality in one variable is an inequality which can be put into the form

ax + b > c

can be written – 4x + (– 2) < – 5.

6x + 1 3(x – 5)

2x + 3 > 4

can be written 6x + 1 – 15.

where a, b, and c are real numbers.

6.1 Solving Linear Inequalities in One Variable

2

The solution set for an inequality can be expressed in two ways.

1. Set-builder notation:

2. Graph on the real line:

{x | x < 3}

Example: Express the solution set of x < 3 in two ways.

1. Set-builder notation:

2. Graph on the real line:

{x | x 4}

Example: Express the solution set of x 4 in two ways.

0 1 2 3 4-4 -3 -2 -1

0 1 2 3 4-4 -3 -2 -1

Open circle indicate that the number is not included in the solution set.

•

°

Closed circle indicate that the number is included in the solution set.


3

A solution of an inequality in one variable is a number which, when substituted for the variable, results in a true inequality.Examples: Are any of the values of x given below solutions of 2x > 5?

2 is not a solution.

2.6 is a solution.

x = 2 2(2) > 5 4 > 5

x = 2.6 2(2.6) > 5 5.2 > 5

The solution set of an inequality is the set of all solutions.

3 is a solution.x = 3 2(3) > 5 6 > 5

x = 1.5 2(1.5) > 5 3 > 5 1.5 is not a solution.False

False

True

True

? ?

? ?

? ?

? ?


4

The graph of a linear inequality in one variable is the graph on the real number line of all solutions of the inequality.


Verbal Phrase Inequality Graph

All real numbers less than 2 x < 2

All real numbers greater than -1 x > -1

All real numbers less than or

equal to 4

x < 4

All real numbers greater than

or equal to -3

x > -3

5

• If a > b and c is a real number, then a > b, a + c > b + c, and a – c > b – c have the same solution set.

Addition and Subtraction Properties

Example: Solve x – 4 > 7.x – 4 > 7 Add 4 to each side of the inequality. + 4+ 4

x > 11Set-builder notation.{x | x > 11}

• If a < b and c is a real number, then a < b, a + c < b + c, and a – c < b – c have the same solution set.


6

Example: Solve 3x 2x + 5.Subtract 2x from each side.3x 2x + 5

x 5

– 2x– 2x

Set-builder notation.{x | x 5}


7

Multiplication and Division Properties

• If c < 0 the inequalities a > b, ac < bc, and < have the same solution set.

c

a

c

b

• If c > 0 the inequalities a > b, ac > bc, and > have the same solution set.

c

ac

b

Example: Solve 4x 12.

x 3

Divide by 4.

4 is greater than 0, so the inequality sign remains the same.

4x 12 )4(

)4(


8

Example: Solve . 43

1 x

4 3

1 x (– 3) (– 3) Multiply by – 3.

– 3 is less than 0, so the inequality sign changes.

12x


9

Example: Solve x + 5 < 9x + 1.

– 8x + 5 < 1 Subtract 9x from both sides.

– 8x < – 4 Subtract 5 from both sides.

x > 2

1 Divide both sides by – 8 and simplify.Inequality sign changes because of division by a negative number.

2

1| xx Solution set in set-builder notation.


10

Example: Solve .45

32

5

4 xx

Subtract from both sides.x5

3425

1x

Add 2 to both sides.65

1x

30x Multiply both sides by 5.

20 30 40 50 60-20 -10 0 10• Solution set as a graph.


11

Example: A cell phone company offers its customers a rate of

$89 per month for 350 minutes, or a rate of $40 per month

plus $0.50 for each minute used.

Solve the inequality 0.50x + 40 89 .

0.50x 49

x 24.5

Subtract 40.

Divide by 0.5.

Let x = the number of minutes used.

The customer can use up to 24.5 minutes per month before the

cost of the second plan exceeds the cost of the first plan.

How many minutes per month can a customer who chooses the

second plan use before the charges exceed those of the first plan?

6.2 Problem Solving

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A compound inequality is formed by joining two inequalities with “and” or “or.”

6.3 Compound Inequalities

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Example: Solve x + 2 < 5 and 2x – 6 > – 8.

x + 2 < 5Solve the first inequality.

The solution set of the “and” compound inequality is the intersection of the two solution sets.

x < 3

{x | x > –1}{x | x < 3}

1|3| xxxx

Subtract 2.

Solution set

Solve the second inequality.2x – 6 > – 8

2x > – 2 Add 6.

Divide by 2. x > – 1Solution set

0 1 2 3 4-4 -3 -2 -1

31| xx

º°


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Example: Solve 11 < 6x + 5 < 29.

6 < 6x < 24 Subtract 5 from each of the three parts.

1 < x < 4 Divide 6 into each of the three parts.

Solution set.

This inequality means 11 < 6x + 5 and 6x + 5 < 29.

When solving compound inequalities, it is possible to work with both inequalities at once.

41| xx


0 1 2 3 4-4 -3 -2 -1

15

Example: Solve .562

18 x

12

12 x

Multiply each part by – 2.24 x

Subtract 6 from each part.

Multiplication by a negative number changes the inequality sign for each part.

0 1 2 3 4-4 -3 -2 -1•• Solution set.


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Example: Solve x + 5 > 6 or 2x < – 4.

Solve the first inequality.

Since the inequalities are joined by “or” the solution set is the union of the solution sets.

Solution set

x + 5 > 6 2x < – 4

Solve the second inequality.

{ x | x > 1}

0 1 2 3 4-4 -3 -2 -1

2|1| xxxx

x > 1 x < – 2

Solution set{ x | x < – 2}

° °


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6.4 Absolute Value and Inequalities

There are 4 types of absolute value inequalities and equivalent inequalities

|x| < a |x| < a |x| > a |x| > a

a x a a x a

x a or x a

x a or x a

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Translating Absolute Value Inequalities

1. The inequality |ax + b| < c is equivalent to -c < ax + b < c

2. The inequality |ax + b| > c is equivalent to ax + b < -c or

ax + b > c

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Example Solve |x - 4| < 3 -3 < x - 4 < 3 1 < x < 7 The solution set is {x| 1 < x < 7} and the

interval is (1, 7)


º º0 1 2 3 4-4 -3 -2 -1

20

Example Solve |4x - 1| < 9 -9 ≤ 4x - 1 ≤ 9 -8 ≤ 4x ≤ 10 -2 ≤ x ≤ 5/2 The interval solution is [-2, 5/2]


0 1 2 3 4-4 -3 -2 -1

21

Solve |x| > a Solve |x + 1| > 2 x + 1 < -2 or x + 1 > 2 x < -3 or x > 1 The solution interval is (-∞, -3) U (1, ∞)


0 1 2 3 4-4 -3 -2 -1º º

22

Solve |x| ≥ a Solve |2x - 8| ≥ 4 2x – 8 ≤ -4 or 2x - 8 ≥ 4 2x ≤ 4 or 2x ≥ 12 x ≤ 2 or x ≥ 6 The solution interval is (-∞, 2] U [ 6, ∞)


• •0 1 2 3 4-4 -3 -2 -1

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6.5 Linear Inequalities A linear inequality in two variables can

be written in any one of these forms: Ax + By < C Ax + By > C Ax + By ≤ C Ax + By ≥ C

An ordered pair (x, y) is a solution of the linear inequality if the inequality is TRUE when x and y are substituted into the inequality.

24

Which ordered pair is a solution of

5x - 2y ≤ 6?A. (0, -3)

B. (5, 5)

C. (1, -2)

D. (3, 3)

6.5 Linear Inequalities

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The graph of a linear inequality is the set of all points in a coordinate plane that represent solutions of the inequality. We represent the boundary line of the

inequality by drawing the function represented in the inequality.


26


The boundary line will be a: Solid line when ≤ and ≥ are used. Dashed line when < and > are

used. Our graph will be shadedshaded on one

side of the boundary line to show where the solutions of the inequality are located.

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Here are some steps to help graph linear inequalities:

1. Graph the boundary line for the inequality. Remember:

≤ and ≥ will use a solid curve. < and > will use a dashed curve.

– Test a point NOT on the boundary line to determine which side of the line includes the solutions. (The origin is always an easy point to test, but make sure your line does not pass through the origin)

If your test point is a solution (makes a TRUE statement), shade THAT side of the boundary line.

If your test points is NOT a solution (makes a FALSE statement), shade the opposite side of the boundary line.

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6.5 Linear Inequalities Graph the inequality x ≤ 4 in a coordinate

plane Decide whether to

use a solid or dashed line.

Use (0, 0) as a test point.

Shade where the solutions will be.

y

x

5

5

-5

-5

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6.5 Linear Inequalities Graph y > x + 2 in a coordinate plane. Sketch the boundary line of the graph. Solid or dashed

line? Use (0, 0) as a

test point. Shade where the

solutions are.

y

x

5

5

-5

-5

30

6.5 Linear Inequalities Graph y > -½x - 2 in a coordinate plane. Sketch the boundary line of the graph. Solid or dashed

line? Use (0, 0) as a

test point. Shade where the

solutions are.

y

x

5

5

-5

-5

31

6.5 Linear Inequalities Graph 3x - 4y > 12 in a coordinate plane. Sketch the boundary line of the graph.

Find the x- and y-intercepts and plot them.

Solid or dashed line?

Use (0, 0) as a test point.

Shade where the solutions are.

y

x

5

5

-5

-5

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Example 4:Using a new Test Point

Graph y < 2/5x in a coordinate plane. Sketch the boundary line of the graph.

Find the x- and y-intercept and plot them. Both are the origin!

• Use the line’s slopeto graph another point.

Solid or dashed line?

Use a test pointOTHER than theorigin.

Shade where the solutions are.

y

x

5

5

-5

-5

Documents

1 Note that the “>” can be replaced by ,