Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 3.3 Linear Inequalities

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Objectives Basic Concepts Symbolic Solutions Numerical and Graphical Solutions An Application

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Basic Concepts An inequality results whenever the equals sign in an equation is replaced with any one of the symbols, or. A solution to an inequality is a value of the variable that makes the statement true. The set of all solutions is called the solution set. Slide 3

Copyright © 2013, 2009, 2005 Pearson Education, Inc. A linear inequality in one variable is an inequality that can be written in the form ax + b > 0, where a 0. (The symbol > may be replaced with, <, or.) LINEAR INEQUALITY IN ONE VARIABLE

Copyright © 2013, 2009, 2005 Pearson Education, Inc. The solution set for ax + b > 0 with a 0 is either {xx k}, where k is the solution to ax + b = 0 and corresponds to the x-intercept for the graph of y = ax + b. Similar statements can be made for the symbols <,, and. SOLUTION SET FOR A LINEAR INEQUALITY

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Let a, b, and c be real numbers. 1.a < b and a + c < b + c are equivalent. (The same number may be added to or subtracted from each side of an inequality.) 2.If c > 0, then a < b and ac < bc are equivalent. (Each side of an inequality may be multiplied or divided by the same positive number.) 3.If c bc are equivalent. (Each side of an equality may be multiplied or divided by the same negative number provided the inequality symbol is reversed.) Similar properties exist for and symbols. PROPERTIES OF INEQUALIES

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve the inequality. 5x – 2 > 9 Solution 5x – 2 > 9 5x – > x > 11 The solution set is.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve the inequality. Solution The solution set is {nn 5}.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Solve the inequality; 9 – 4x x 6. Solution Next divide each side by 3. As when we are dividing by a negative number, Property 3 requires reversing the inequality by changing to. 9 – 4x x – 6 9 – 4x – 9 x – 6 – 9 4x x – 15 4x + x x – 15 + x 3x – 15 The solution set is {xx 5}.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Use the table to find the solution set to each equation or inequality. a.b.c. x a. The expression equals 0 when x = 2. Thus the solution set is {xx = 2}.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Use the table to find the solution set to each equation or inequality. a.b.c. x b. The expression is positive when x < 2. Thus the solution set is {xx < 2}.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Use the table to find the solution set to each equation or inequality. a.b.c. x c. The expression is negative when x > 2. Thus the solution set is {xx > 2}.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Use the graph to find the solution set to each equation or inequality. a. The graph of crosses the x-axis at x = 2. Thus the solution set is

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Use the graph to find the solution set to each equation or inequality. b. The graph is above the x-axis when x < 2. Thus the solution set is {xx < 2}.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Use the graph to find the solution set to each equation or inequality. c. The graph is above the x-axis when x > 2. Thus the solution set is {xx > 2}.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Use a graph to solve 7 – 4x x – 8. Solution The graph of y 1 = 7 – 4x and y 2 = x – 8 intersect at the point (3, 5). The graph of y 1 is below the graph of y 2 when x > 3. Thus 7 – 4x x – 8 is satisfied when x 3. Therefore the solution set is {xx 3}.