© William James Calhoun, : Solving Inequalities by Using Multiplication and Division OBJECTIVE: You will solve inequalities by using multiplication and division. The rules for multiplying and dividing both sides of an inequality are essentially the same as for equations. There is one key difference you must remember to do: WHENEVER YOU MULTIPLY OR DIVIDE BOTH SIDES BY A NEGATIVE NUMBER, YOU MUST SWITCH THE DIRECTION OF THE INEQUALITY. 2 < 3 3(2) < 3(3) 6 < 9 -6 <  (3) < -3  (3) -2 < -1 Both statements are still true, so multiplying and dividing by a positive number did not affect the inequality symbol. 2 < 3 -3(2) < -3(3) -6 < <  (-3) < -3  (-3) 2 < 1 Both statements are now false. In order to get that back to true statement, the inequality sign must be flipped. -6 > -92 > 1

© William James Calhoun, : Solving Inequalities by Using Multiplication and Division EXAMPLE 1: Solve the inequality. What is the variable? LeLe What is on the same side? 1.5 How are they combined? division To undo… multiply both sides by 1.5 Now, since both sides were multiplied by something, pause and ask yourself, “Did I multiply both sides by a negative number?” The answer is no, so do not change the inequality sign. EXAMPLE 2: Solve What is the variable? x What is on the same side? 12 How are they combined? division To undo… multiply both sides by 12 Now, since both sides were multiplied by something, pause and ask yourself, “Did I multiply both sides by a negative number?” The answer is no, so do not change the inequality sign. {L e | L e  10.5} {x | x  18}

© William James Calhoun, : Solving Inequalities by Using Multiplication and Division EXAMPLE 3: Solve -3w > 27. What is the variable? w What is on the same side? negative 3 How are they combined? multiplication To undo… divide both sides by -3 Now, since both sides were divided by something, pause and ask yourself, “Did I divide both sides by a negative number?” The answer is yes, so change the inequality sign. -3w > w < -9 {w | w < -9}

© William James Calhoun, 2001 EXAMPLE 4: Angelica Moreno is a sales representative for an appliance distributor. She needs more than $5000 in weekly sales of a particular TV model to qualify for a sales competition to win a trip to the Bahamas. If the TVs sell for $250 each, how many TVs will Ms. Moreno have to sell to qualify? Moreno needs to have sales figures that total more than $5000 to win. Her sales figures are based on how many she sells times the number she sells. $250 times the number of TVs she sells gives her sales figure. Let T be the number of TVs she sells, giving the inequality: 250(T) >5000 Rewrite and solve. 250 T > T > : Solving Inequalities by Using Multiplication and Division Moreno needs to sell more than 20 TVs to win the trip. Note, selling 20 exactly will not earn her the trip, it must be 21 or more.

© William James Calhoun, : Solving Inequalities by Using Multiplication and Division HOMEWORK Page 396 # odd