chapter p prerequisites: fundamental concepts of algebra 1 copyright © 2014, 2010, 2007 pearson...

Chapter PPrerequisites: Fundamental Concepts of Algebra 1

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

P.9 Linear Inequalitiesand Absolute ValueInequalities


• Use interval notation.• Find intersections and unions of intervals.• Solve linear inequalities.• Solve compound inequalities.• Solve absolute value inequalities.

Objectives:


Solving an Inequality

Solving an inequality is the process of finding the set of numbers that make the inequality a true statement. These numbers are called the solutions of the inequality and we say that they satisfy the inequality. The set of all solutions is called the solution set of the inequality.


Interval Notation

The open interval (a,b) represents the set of real numbers between, but not including, a and b.

The closed interval [a,b] represents the set of real numbers between, and including, a and b.

( , )a b x a x b

[ , ]a b x a x b


A half-open, or half-closed interval is (a, b], consisting of all real numbers x for which a < x < b.

( ]a b


A half-open, or half-closed interval is [a, b), consisting of all real numbers x for which a < x < b.

[ )a b


[a


(a


Interval Notation (continued)

The infinite interval represents the set of real numbers that are greater than a.

The infinite interval represents the set of real numbers that are less than or equal to b.

( , )a

( , )a x x a

( , ]b x x b

( , ]b


Parentheses and Brackets in Interval Notation

Parentheses indicate endpoints that are not included in an interval. Square brackets indicate endpoints that are included in an interval. Parentheses are always used with or .


Example: Using Interval Notation

Express the interval in set-builder notation and graph:

[1, 3.5]

Express the interval in set-builder notation and graph:

1 3.5x x

( , 1)

1x x


3 2,

[ )-3 20

Write the inequality -3 < x < 2 using

interval notation. Illustrate the inequality

using a real number line.


Sets and Intervals


Sets and IntervalsA set is a collection of objects, and these objects are called the elements of the set. If S is a set, the notation a S means that a is an element of S, and b S means that b is not an element of S.

For example, if Z represents the set of integers, then –3 Z but Z.

Some sets can be described by listing their elements within braces. For instance, the set A that consists of all positive integers less than 7 can be written as

A = {1, 2, 3, 4, 5, 6}


Sets and Intervals

We could also write A in set-builder notation as

A = {x | x is an integer and 0 < x < 7}

which is read “A is the set of all x such that x is an integer and 0 < x < 7.”

If S and T are sets, then their union S T is the set that consists of all elements that are in S or T (or in both). The intersection of S and T is the set S T consisting of all elements that are in both S and T.


Sets and Intervals

In other words, S T is the common part of S and T. The empty set, denoted by Ø, is the set that contains no element.


Example 4 – Union and Intersection of Sets

If S = {1, 2, 3, 4, 5}, T = {4, 5, 6, 7}, and V = {6, 7, 8}, find the sets S T, S T, and S V.

Solution:

S T = {1, 2, 3, 4, 5, 6, 7}

S T = {4, 5}

S V = Ø

All elements in S or T

Elements common to both S and T

S and V have no element in common


Finding Intersections and Unions of Two Intervals

1. Graph each interval on a number line.

2. a. To find the intersection, take the portion of the

number line that the two graphs have in common.

b. To find the union, take the portion of the number

line representing the total collection of numbers

in the two graphs.


Example: Finding Intersections and Unions of Intervals

Use graphs to find the set:

Graph of [1,3]:

Graph of (2,6):

Numbers in both [1,3] and (2,6):

Thus,

[1,3] (2,6).

[1,3] (2,6) (2,3].


Solving Linear Inequalities in One Variable

A linear inequality in x can be written in one of the following forms :

In general, when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol is reversed.

0a 0ax b 0ax b

0ax b 0ax b


Example: Solving a Linear Inequality

Solve and graph the solution set on a number line:

2 3 5. x

2 3 5x

3 3x

3 33 3x

1x

The solution set is . 1x x

The number line graph is:

The interval notation for thissolution set is .[ 1, )


Example: Solving a Compound Inequality

Solve and graph the solution set on a number line:1 2 3 11. x

Our goal is to isolate x in the middle.

2 2 8x 1 4x

In interval notation, the solution is [-1,4).

In set-builder notation, the solution set is 1 4x x

1 2 3 11x

The number line graph looks like


Solving an Absolute Value Inequality

If u is an algebraic expression and c is a positive number,

1. The solutions of are the numbers that satisfy

2. The solutions of are the numbers that satisfy

or

These rules are valid if is replaced by and

is replaced by

u c

u c. c u c

u c .u c

.


Example: Solving an Absolute Value Inequality

Solve and graph the solution set on a number line:

18 6 3 . x

We begin by expressing the inequality with the absolute value expression on the left side:

6 3 18x

We rewrite the inequality without absolute value bars. means or 6 3 18x 6 3 18x 6 3 18. x


Example: Solving an Absolute Value Inequality (continued)

We solve these inequalities separately:6 3 18x

3 24x 3 243 3x

8x

6 3 18x 3 12x 3 123 3x

4x

The solution set is 4 or 8x x x

The number line graph looks like

chapter p prerequisites: fundamental concepts of algebra 1 copyright © 2014, 2010, 2007 pearson...

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