Solving Inequalities By Multiplying Or Dividing

SolvingInequalitiesSolvingInequalitiesby byMultiplyingor DividingMultiplyingor DividingWarm UpLesson PresentationLesson QuizHoltAlgebra1 AlgebraHoltHoltMcDougalMcDougalAlgebra11

Solving Inequalities byMultiplying or DividingWarm UpSolve each equation.1. –5a 30 –62.3.4.Graph each inequality.5. x –106. x –3Holt McDougal Algebra 1–10

Solving Inequalities byMultiplying or DividingObjectivesSolve one-step inequalities by usingmultiplication.Solve one-step inequalities by using division.Holt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingRemember, solving inequalities is similar tosolving equations. To solve an inequality thatcontains multiplication or division, undo theoperation by dividing or multiplying both sides ofthe inequality by the same number.The following rules show the properties ofinequality for multiplying or dividing by apositive number. The rules for multiplying ordividing by a negative number appear later inthis lesson.Holt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingHolt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingExample 1A: Multiplying or Dividing by a PositiveNumberSolve the inequality and graph the solutions.7x –427x –42Since x is multiplied by 7, divide bothsides by 7 to undo the multiplication. 1x –6x –6–10 –8 –6 –4 –2 0Holt McDougal Algebra 12468 10

Solving Inequalities byMultiplying or DividingExample 1B: Multiplying or Dividing by a PositiveNumberSolve the inequality and graph the solutions.Since m is divided by 3, multiply bothsides by 3 to undo the division.3(2.4) 37.2 m (or m 7.2)02468 10 12 14 16 18 20Holt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingExample 1C: Multiplying or Dividing by a PositiveNumberSolve the inequality and graph the solutions.Since r is multiplied byr 1602468 10 12 14 16 18 20Holt McDougal Algebra 1,multiply both sides by thereciprocal of .

Solving Inequalities byMultiplying or DividingCheck It Out! Example 1aSolve the inequality and graph the solutions.4k 24Since k is multiplied by 4, divideboth sides by 4.k 602468 10 12 14 16 18 20Holt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingCheck It Out! Example 1bSolve the inequality and graph the solutions.–50 5qSince q is multiplied by 5, divideboth sides by 5.–10 q–15–10–5Holt McDougal Algebra 10515

Solving Inequalities byMultiplying or DividingCheck It Out! Example 1cSolve the inequality and graph the solutions.Since g is multiplied bymultiply both sides by thereciprocal of .g 3636152025Holt McDougal Algebra 130,3540

Solving Inequalities byMultiplying or DividingIf you multiply or divide both sides of aninequality by a negative number, the resultinginequality is not a true statement. You need toreverse the inequality symbol to make thestatement true.Holt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingThis means there is another set of propertiesof inequality for multiplying or dividing by anegative number.Holt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingHolt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingCaution!Do not change the direction of the inequalitysymbol just because you see a negativesign. For example, you do not change thesymbol when solving 4x –24.Holt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingExample 2A: Multiplying or Dividing by a NegativeNumberSolve the inequality and graph the solutions.–12x 84Since x is multiplied by –12, divideboth sides by –12. Change to .x –7–7–14 –12 –10 –8 –6 –4 –2Holt McDougal Algebra 10246

Solving Inequalities byMultiplying or DividingExample 2B: Multiplying or Dividing by a NegativeNumberSolve the inequality and graph the solutions.Since x is divided by –3, multiplyboth sides by –3. Change to .24 x (or x 24)10 12 14 16 18 20 22 24 26 28 30Holt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingCheck It Out! Example 2Solve each inequality and graph the solutions.a. 10 –x–1(10) –1(–x)Multiply both sides by –1 to make xpositive. Change to .–10 x–10 –8 –6 –4 –2 02468 10b. 4.25 –0.25hSince h is multiplied by –0.25, divideboth sides by –0.25. Change to .–17 hHolt McDougal Algebra 1–17–20 –16 –12 –8 –4 048 12 16 20

Solving Inequalities byMultiplying or DividingExample 3: ApplicationJill has a 20 gift card to an art supply storewhere 4 oz tubes of paint are 4.30 each aftertax. What are the possible numbers of tubesthat Jill can buy?Let p represent the number of tubes of paint that Jillcan buy. 4.30times4.30 Holt McDougal Algebra 1number of tubesis at most 20.00.p 20.00

Solving Inequalities byMultiplying or DividingExample 3 Continued4.30p 20.00Since p is multiplied by 4.30,divide both sides by 4.30. Thesymbol does not change.p 4.65 Since Jill can buy only whole numbers of tubes,she can buy 0, 1, 2, 3, or 4 tubes of paint.Holt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingCheck It Out! Example 3A pitcher holds 128 ounces of juice. What arethe possible numbers of 10-ounce servings thatone pitcher can fill?Let x represent the number of servings of juice thepitcher can contain.10 oz10timesnumber ofservingsis at most128 oz x 128Holt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingCheck It Out! Example 3 Continued10x 128Since x is multiplied by 10, divide bothsides by 10.The symbol does not change.x 12.8The pitcher can fill 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10, 11, or 12 servings.Holt McDougal Algebra 1

Solving Inequalities byMultiplying or DividingLesson QuizSolve each inequality and graph the solutions.1. 8x –24 x –32. –5x 30x –63.4.x 6x 205. A soccer coach plans to order more shirts forher team. Each shirt costs 9.85. She has 77left in her uniform budget. What are thepossible number of shirts she can buy?0, 1, 2, 3, 4, 5, 6, or 7 shirtsHolt McDougal Algebra 1

Multiplying or Dividing Since x is divided by –3, multiply both sides by –3. Change to . 10 12 14 16 18 20 22 24 26 28 30 Example 2B: Multiplying or Dividing by a Negative Number Solve the inequality and graph the solutions. 24 x (or x 24)