Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1 1 Section 2.7 Solving Linear Inequalities Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

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3 Any inequality in the form is called a linear inequality in one variable. The inequality symbol may be < less than > greater than ≤ less than or equal to ≥ greater than or equal to Linear Inequalities in One Variable

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 4 Graphs of Inequalities

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 5 Linear Inequalities EXAMPLE SOLUTION Graph the solution of each inequality on a number line and then express the solutions in set-builder notation and interval notation. -4 (

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 6 Linear Inequalities EXAMPLE SOLUTION Graph the solution of each inequality on a number line and then express the solutions in set-builder notation and interval notation. -4 (

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 7 Linear Inequalities CONTINUED ) [ -27

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 8 Linear Inequalities CONTINUED ) [ -27

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 9 Objective #1: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 10 Objective #1: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 11 Objective #1: Examples

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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 14 Interval Notation

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 15 Interval Notation

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 16 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 16 Objective #2: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 17 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 17 Objective #2: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 18 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 18 Objective #2: Examples

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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 21 Inequalities Properties of Inequalities PropertyThe Property in WordsExample The Addition Property of Inequality If a < b, then a + c < b + c If a < b, then a – c < b - c If the same quantity is added to or subtracted from both sides of an inequality, the resulting inequality is equivalent to the original one. 2x + 3 < 7 Subtract 3: 2x + 3 – 3 < 7 – 3 Simplify: 2x < 4

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 22 Inequalities Properties of Inequalities PropertyThe Property in WordsExample The Positive Multiplication Property of Inequality If a < b and c is positive, then ac < bc If a < b and c is positive, then If we multiply or divide both sides of an inequality by the same positive quantity, the resulting inequality is equivalent to the original one. 2x < 4 Divide by 2: Simplify: x < 2 CONTINUED

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 23 Inequalities Properties of Inequalities PropertyThe Property in WordsExample The Negative Multiplication Property If a bc If a < b and c is negative, then If we multiply or divide both sides of an inequality by the same negative quantity and reverse the direction of the inequality symbol, the resulting inequality is equivalent to the original one. CONTINUED Remember that when you multiply both sides of an inequality by a negative, you must reverse the direction of the inequality symbol.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 24 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 24 Objective #3: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 25 Objective #3: Examples

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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 27 Inequalities Solving a Linear Inequality 1) Simplify the algebraic expression on each side. 2) Use the addition property of inequality to collect all the variable terms on one side and all the constant terms on the other side. 3) Use the multiplication property of inequality to isolate the variable and solve. Reverse the sense of the inequality when multiplying or dividing both sides by a negative number. 4) Express the solution set in set-builder or interval notation and graph the solution set on a number line.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 28 EXAMPLE: Solve 2(x + 3) ≤ 4x + 12 Linear Inequalities

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 29 EXAMPLE: Solve 2(x + 3) ≤ 4x + 12 Linear Inequalities

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 30 Linear Inequalities EXAMPLE SOLUTION Solve the linear inequality. Then graph the solution set on a number line. Distribute Add 5x to both sides Add 1 to both sides 1) Simplify each side. 2) Collect variable terms on one side and constant terms on the other side.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 31 Linear Inequalities 3) Isolate the variable and solve [ Divide both sides by 8 CONTINUED 4) Express the solution set in set-builder or interval notation and graph the set on a number line. {x|x 2} = [2, )

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 32 Linear Inequalities EXAMPLE SOLUTION Solve the linear inequality. Then graph the solution set on a number line. Distribute First we need to eliminate the denominators. Multiply by LCD = 10

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 33 Linear Inequalities Add x to both sides Subtract 10 from both sides 1) Simplify each side. Because each side is already simplified, we can skip this step. 2) Collect variable terms on one side and constant terms on the other side. CONTINUED 1 2 1

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 34 Linear Inequalities CONTINUED 3) Isolate the variable and solve [ Divide both sides by 4 4) Express the solution set in set-builder or interval notation and graph the set on a number line. {x|x -2} = [-2, )

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 35 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 35 Objective #4: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 36 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 36 Objective #4: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 37 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 37 Objective #4: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 38 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 38 Objective #4: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 39 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 39 Objective #4: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 40 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 40 Objective #4: Examples

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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 42 No Solution or Infinitely Many Solutions Some inequalities have either no solution or have infinitely many solutions. When you attempt to solve such inequalities, all the variables are eliminated and you either get a sentence that is true or false. If the variables are all eliminated at once and you are left with a true sentence, the original inequality is an identity and is true for all x. If the variables are all eliminated at once and you are left with a false sentence, the original inequality has no solution.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 43 Linear Inequalities EXAMPLE SOLUTION Solve the linear inequality. Distribute Since the result is a false statement, there is no solution. 5x < 5(x – 3) 5x < 5x – 15 Subtract 5x from both sides0 < – 15

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 44 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 44 Objective #5: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 45 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 45 Objective #5: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 46 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 46 Objective #5: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 47 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 47 Objective #5: Examples

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Copyright © 2013, 2009, 2006 Pearson Education, Inc. 49 Linear Inequalities EXAMPLE SOLUTION You are choosing between two long-distance telephone plans. Plan A has a monthly fee of $15 with a charge of $0.08 per minute for all long-distance calls. Plan B has a monthly fee of $3 with a charge of $0.12 per minute for all long-distance calls. How many minutes of long-distance calls in a month make plan A the better deal? 1) Let x represent one of the quantities. We are looking for the number of minutes of long-distance calls that make Plan A the better deal. Thus, Let x = the number of minutes of long-distance phone calls.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 50 Linear Inequalities 2) Represent other quantities in terms of x. We are not asked to find another quantity, so we can skip this step. 3) Write an inequality in x that describes the conditions. Plan A is a better deal than Plan B if the monthly cost of Plan A is less than the monthly cost of Plan B. The inequality that represents this is the following, with the information for Plan A on the left side and the information for Plan B on the right side of the inequality. CONTINUED x < x Plan A is less than Plan B.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 51 Linear Inequalities 4) Solve the inequality and answer the question. Subtract 3 from both sides CONTINUED x < x x < 0.12x Subtract 0.08x from both sides15 < 0.04x Divide both sides by < x Therefore, Plan A costs less than Plan B when x > 375 long- distance minutes per month.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 52 Linear Inequalities 5) Check the proposed solution in the original wording of the problem. One way to do this is to take a long-distance number of minutes greater than 375 per month to see if Plan A is the better deal. Suppose that the number of long-distance minutes we use is 450 in a month. CONTINUED Cost for Plan A = 15 + (450)(0.08) = $51 Plan A has a lower monthly cost, making Plan A the better deal. Cost for Plan B = 3 + (450)(0.12) = $57

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 53 Linear Inequalities In summary… Solving a linear inequality is similar to solving a linear equation except for two important differences: When you solve an inequality, your solution is usually an interval, not a point. and When you multiply both sides of an inequality by a negative number, you must remember to reverse the direction of the inequality sign.

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 54 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 54 Objective #6: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 55 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 55 Objective #6: Examples

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 56 Copyright © 2013, 2009, 2006 Pearson Education, Inc. 56 Objective #6: Examples CONTINUED