Chapter P Prerequisites: Fundamental Concepts of Algebra 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 P.9 Linear Inequalities and Absolute Value Inequalities

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Use interval notation. Find intersections and unions of intervals. Solve linear inequalities. Solve compound inequalities. Solve absolute value inequalities. Objectives:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 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.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 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.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 A half-open, or half-closed interval is (a, b], consisting of all real numbers x for which a < x < b. (] ab

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 A half-open, or half-closed interval is [a, b), consisting of all real numbers x for which a < x < b. [) ab

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 [ a

8 ( a

9 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.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 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.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 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:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 [) Write the inequality -3 < x < 2 using interval notation. Illustrate the inequality using a real number line.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Sets and Intervals

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Sets and Intervals A 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}

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 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.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 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.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 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

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 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.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 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,

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 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.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Solving a Linear Inequality Solve and graph the solution set on a number line: The solution set is. The number line graph is: The interval notation for this solution set is.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Solving a Compound Inequality Solve and graph the solution set on a number line: Our goal is to isolate x in the middle. In interval notation, the solution is [-1,4). In set-builder notation, the solution set is The number line graph looks like

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 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

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Solving an Absolute Value Inequality Solve and graph the solution set on a number line: We begin by expressing the inequality with the absolute value expression on the left side: We rewrite the inequality without absolute value bars. means or

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Solving an Absolute Value Inequality (continued) We solve these inequalities separately: The solution set is The number line graph looks like