Solving inequalities Using Multiplication or Division Section 3-3

Goals Goal To use multiplication or division to solve inequalities. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary None

Solving Inequalities Using Multiplication or Division Remember, solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division, undo the operation by dividing or multiplying both sides of the inequality by the same number. The rules on the next slide show the properties of inequality for multiplying or dividing by a positive number. The rules for multiplying or dividing by a negative number appear later in this lesson.

Properties of Inequality for Multiplication and Division by Positive Numbers

When graphing an inequality on a number line, an open circle means that the point is not part of the solution and a closed circle means that the point is part of the solution. Remember!

Solve the inequality and graph the solutions. 7x > –42 > 1x > –6 Since x is multiplied by 7, divide both sides by 7 to undo the multiplication. x > –6 –10 –8 –6–4 – Example: Multiplying or Dividing by a Positive Number

3(2.4) ≤ ≤ m(or m ≥ 7.2) Since m is divided by 3, multiply both sides by 3 to undo the division Solve the inequality and graph the solutions. Example: Multiplying or Dividing by a Positive Number 7.2 |

r < Since r is multiplied by, multiply both sides by the reciprocal of. Solve the inequality and graph the solutions. Example: Multiplying or Dividing by a Positive Number

Solve the inequality and graph the solutions. 4k > 24 k > Since k is multiplied by 4, divide both sides by 4. Your Turn:

–50 ≥ 5q –10 ≥ q (or q ≤ -10) Since q is multiplied by 5, divide both sides by 5. Solve the inequality and graph the solutions. 5–50 –10–15 15 Your Turn:

g > 36 Since g is multiplied by, multiply both sides by the reciprocal of Solve the inequality and graph the solutions. Your Turn:

Example: Multiplying or Dividing by a Negative Number If you multiply or divide both sides of an inequality by a negative number, the resulting inequality is not a true statement. You need to reverse the inequality symbol to make the statement true. This is the main difference between solving inequalities and solving equations. This means there is another set of properties of inequality for multiplying or dividing by a negative number.

Properties of Inequality for Multiplication and Division by Negative Numbers

Caution! Do not change the direction of the inequality symbol just because you see a negative sign. For example, you do not change the symbol when solving 4x < –24.

Solve the inequality and graph the solutions. –12x > 84 x < –7 Since x is multiplied by –12, divide both sides by –12. Change > to <. –10 –8–8 –6–6–4–4 –2– –12–14 –7 Example: Multiplying or Dividing by a Negative Number

Since x is divided by –3, multiply both sides by –3. Change to Solve the inequality and graph the solutions. 24  x(or x  24) Example: Multiplying or Dividing by a Positive Number

Solve each inequality and graph the solutions. a. 10 ≥ –x –1(10) ≤ –1(–x) –10 ≤ x (or x ≥ -10) Multiply both sides by –1 to make x positive. Change  to . b > –0.25h –17 < h (or h > -17) Since h is multiplied by –0.25, divide both sides by –0.25. Change > to <. –20 –16 –12–8 – –17 –10 –8 –6–4 – Your Turn:

$4.30 times number of tubes is at most $ p ≤ Jill has a $20 gift card to an art supply store where 4 oz tubes of paint are $4.30 each after tax. What are the possible numbers of tubes that Jill can buy? Let p represent the number of tubes of paint that Jill can buy. Example: Application

4.30p ≤ p ≤ 4.65… Since p is multiplied by 4.30, divide both sides by The symbol does not change. Since Jill can buy only whole numbers of tubes, she can buy 0, 1, 2, 3, or 4 tubes of paint. Example: Continued

A pitcher holds 128 ounces of juice. What are the possible numbers of 10-ounce servings that one pitcher can fill? 10 oz times number of servings is at most 128 oz 10 x ≤ 128 Let x represent the number of servings of juice the pitcher can contain. Your Turn:

10x ≤ 128 Since x is multiplied by 10, divide both sides by 10. The symbol does not change. x ≤ 12.8 The pitcher can fill 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 servings. Your Turn: Continued

Joke Time What’s a cow’s favorite painting? The Moona Lisa What does the tooth fairy give for half a tooth? Nothing. She wants the tooth, the whole tooth, and nothing but the tooth! What do you get if you take a native Alaskan and divide its circumference by its diameter? Eskimo pi

Assignment 3-3 Exercises Pg. 194 – 196: #8 – 62 even