Slide 1.6- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

OBJECTIVES

Linear Inequalities

Learn the vocabulary for discussing inequalities.

Learn to solve and graph linear inequalities.

Learn to solve and graph a combined inequality.

Learn to solve and graph an inequality involving the reciprocal of a linear expression.

SECTION 1.6

1

2

3

4


INEQUALITIES

EquationReplace =

byInequality

x = 5 < x < 5

3x + 2 = 14 ≤ 3x + 2 ≤ 14

5x + 7 = 3x + 23 > 5x + 7 > 3x + 23

x2 = 0 ≥ x2 ≥ 0


DefinitionsAn inequality is a statement that one algebraic expression is less than, or is less than or equal to, another algebraic expression.

The domain of a variable in an inequality is the set of all real numbers for which both sides of the inequality are defined.

The solutions of the inequality are the real numbers that result in a true statement when those numbers are substituted for the variable in the inequality.


DefinitionsTo solve an inequality means to find all solutions of the inequality–that is, the solution set.

The graph of the inequality x < 5 is the interval (–∞, 5) and is shown here.

The solution sets are intervals, and we frequently graph the solutions sets for inequalities in one variable on a number line.

x < 5, or (–∞, 5)

)5


Definitions

A conditional inequality such as x < 5 has in its domain at least one solution and at least one number that is not a solution.

An inconsistent inequality is one in which no real number satisfies it.

An identity is an inequality that is satisfied by every real number in the domain.


THE NONNEGATIVE IDENTITY

for any real number x.

x2 0

Because x2 = x•x is the product of either (1) two positive factors, (2) two negative factors, or (3) two zero factors, x2 is always either a positive number or zero. That is, x2 is never negative, or is nonneagtive.


EQUIVALENT INEQUALITIES

1. Simplifying one or both sides of an inequality by combining like terms and eliminating parentheses

2. Adding or subtracting the same expression on both sides of the inequality


Signof C

Inequality Sense Example

A < B < 3x < 12

positive A•C < B •C Unchanged

positive Unchanged

negative A•C > B •C Reversed

negative Reversed

1

33x 1

312

3x

3

12

3

1

33x

1

312

3x

3

12

3

A

C

B

C

A

C

B

C


Linear Inequalities

A linear inequality in one variable is an inequality that is equivalent to one of the forms

ax + b < 0 or ax + b ≤ 0,

where a and b represent real numbers and a ≠ 0.


EXAMPLE 1 Solving and Graphing Linear Inequalities

Solve each inequality and graph its solution set.

The solution set is {x|x < 1}, or (–∞, 1).

a. 7x 11 2 x 3 b. 8 3x 2

Solution7x 11 2 x 3

7x 1111 2 x 3 11

7x 2x 5

7x 2x 2x 5 2x

5x 5

5x

5

5

5x 1

)0 1–1


EXAMPLE 1 Solving and Graphing Linear Inequalities

The solution set is {x|x ≥ 2}, or [2, ∞).

Solution continued

8 3x 2

8 3x 8 2 8

3x 6

3x

3

6

3x 2

[1 20

b. 8 3x 2


EXAMPLE 2Calculating the Results of the Bermuda Triangle Experiment

In the introduction to this section, we discussed an experiment to test the reliability of compass settings and flight by automatic pilot along one edge of the Bermuda Triangle. The plane is 150 miles along its path from Miami to Bermuda, cruising at 300 miles per hour, when it notifies the tower that it is now set on automatic pilot.

The entire trip is 1035 miles, and we want to determine how much time we should let pass before we become concerned that the plane has encountered trouble.


EXAMPLE 2

Let t = time elapsed since plane on autopilot300t = distance plane flown in t hours150 + 300t = plane’s distance from Miami after

t hours

Solution

Calculating the Results of the Bermuda Triangle Experiment

Plane’s distance from Miami

Distance from Miami to Bermuda

≥

150 300t 1035

150 300t 150 1035 150


EXAMPLE 2

Solution continued

Calculating the Results of the Bermuda Triangle Experiment

300t 885

300t

300

885

300t 2.95

Since 2.95 is roughly 3 hours, the tower will suspect trouble if the plane has not arrived in 3 hours.


EXAMPLE 3 Solving & Graphing a Compound Inequality

Solve the inequality 5x 2x 3 9 graph its solution set.

and

Solution

First, solve the inequalities separately.

5 2x 3

5 3 2x 3 3

8

2

2x

2 4 x

2x 3 9

2x 3 3 9 3

2x

2

6

2x 3


EXAMPLE 3 Solving & Graphing a Compound Inequality

We can write this as {x| – 4 < x ≤ 3}. In interval notation we write (– 4, 3].

Solution continued

The solution to the original inequalities consists of all real numbers x such that – 4 < x and x ≤ 3.

]31

(–1 0 4– 4– 5 2–2–3


THE RECIPROCAL SIGN PROPERTY

If x ≠ 0, x and 1

xare either both positive or

negative. In symbols, if x > 0, then 1

x 0

and if x < 0, then1

x 0.


EXAMPLE 4Solving and Graphing an Inequality by Using the Reciprocal Sign Property

Solve and graph 3x 12 1 0.

Solution3x 12 1 0

1

3x 12 0

3x 12 0

3x 12

3x

3

12

3x 4

The solution set is {x| x >4}, or in interval notation (4, ∞).

[1 20 4 53


EXAMPLE 5Finding the Interval of Values for a Linear Expression

If –2 < x < 5, find real numbers a and b so that a < 3x – 1 < b.

Solution 2 x 5

3 2 3x 3 5 6 3x 15

6 1 3x 1 15 1

7 3x 1 14

We have a = –7 and b = 14.


EXAMPLE 6Finding a Fahrenheit Temperature from a Celsius Range

The weather in London is predicted to range between 10º and 20º Celsius during the three-week period you will be working there. To decide what kind of clothes to bring, you want to convert the temperature range to Fahrenheit temperatures. The formula for converting Celsius temperature C to Fahrenheit temperature F is

temperatures might you find in London during your stay there?

F 9

5C 32. What range of Fahrenheit



Let C = temperature in Celsius degrees.Solution

9

510 9

5C

9

520

18 9

5C 36

18 32 9

5C 32 36 32

For the three weeks under consideration10 ≤ C ≤ 20.



Solution continued

50 9

5C 32 68

50 F 68

So, the temperature range from 10º to 20º Celsius corresponds to a range from 50º to 68º Fahrenheit.

Slide 1.6- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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