Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 2 Linear Functions and Equations

2 Copyright © 2014, 2010, 2006 Pearson Education, Inc. Linear Inequalities ♦ Understand basic terminology related to inequalities ♦ Solve linear inequalities symbolically ♦ Solve linear inequalities graphically and numerically ♦ Solve compound inequalities 2.3

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 3 Terminology Related to Inequalities Inequalities result whenever the equals sign in an equation is replaced with any one of the symbols ≤, ≥,. Examples of inequalities include: x + 15 < 9x – 1 x 2 – 2x + 1 ≥ 2x z + 5 > 0 xy + x 2 ≤ y 3 + x > 1

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 4 Linear Inequality in One Variable 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 by ≤, ≥,.) Examples of linear inequalities in one variable: 3x – 4 < 07x + 5 ≥ x x + 6 > 237x + 2 ≤ –3x + 6

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 5 Linear Inequality in One Variable Using techniques from algebra, we can transform these inequalities into one of the forms ax + b > 0, ax + b ≥ 0, ax + b < 0, ax + b ≤ 0,. For example, by subtracting x from each side of 7x + 5 ≥ x,, we obtain the equivalent inequality 6x + 5 ≥ 0. If an inequality is not a linear inequality, it is called a nonlinear inequality.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 6 Properties of Inequalities 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.)

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 7 Properties of Inequalities 3.If c bc are equivalent. (Each side of an inequality may be multiplied or divided by the same negative number provided the inequality symbol is reversed.) Replacing with ≥ results in similar properties.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 8 Review of Interval Notation 3 ≤ x ≤ 5 is written as [3, 5] 3 < x < 5 is written as (3, 5) A bracket [ or ] is used when the endpoint is included A parenthesis ( or ) is used when the endpoint is not included. x ≥ 2 is written [2, ∞) x < 2 is written (–∞,2)

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 9 Solve each inequality. Write the solution set in set-builder and interval notation. Example: Solving a linear inequality symbolically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 10 Solution (a) Property 3, multiply both sides by –3 Set-builder notation: Interval notation: Example: Solving a linear inequality symbolically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 11 (b) Begin with distributive property Set-builder notation: Interval notation: Example: Solving a linear inequality symbolically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 12 Graph by hand. Use the graph to solve the linear inequality Example: Solving a linear inequality graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 13 Solution The graphs intersect at the point (2, 3). The graph of is above the graph of y 2 = 2x – 1 to the left of (2, 3), or when x < 2. Solution is {x | x < 2}, or (–∞, 2). Example: Solving a linear inequality graphically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 14 x-Intercept Method If a linear inequality can be written as y 1 > 0, where > may be replaced by ≥, ≤, or 0, graph y 1 and find the x-intercept. The solution set includes x-values where the graph of y 1 is above the x- axis.

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 15 Solve numerically. Make a table of Y 1 = 3(6 – X) + 5 – 2X Boundary lies between x = 4 and x = 5 Solution Example: Solving a linear inequality with test values

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 16 Change increment from 1 to 0.1 Boundary is x = 4.6 Test values of x = 4.7, 4.8, 4.9 indicate when x > 4.6, y 1 < 0. Solution set is {x | x > 4.6}. Example: Solving a linear inequality with test values

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 17 Compound Inequalities A compound inequality occurs when two inequalities are connected by the word and or or. When the word and connects two inequalities, the two inequalities can sometimes be written as a three-part inequality. x ≥ 40 and x ≤ 70 can be written 40 ≤ x ≤ 70

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 18 Solve the inequality. Write the solution set in set- builder and interval notation. Solution set is {x | –1 < x < 4 or [–1, 4) Solution Example: Solving a three-part inequality symbolically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 19 (b) Multiply each part by 4 Solution set is Example: Solving a three-part inequality symbolically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 20 Solve the linear inequality symbolically. Express the solution set using interval notation. Solution The parts of this compound inequality can be solved simultaneously. Example: Solving inequalities symbolically

Copyright © 2014, 2010, 2006 Pearson Education, Inc. 21 Example: Solving inequalities symbolically The solution set is