Skills You Need 6n = = -7v = 10 = 48 6 n = = v (-2) n = -20 (6) t = 288

2.10 Using Multiplication & Division with Inequalities……..something new…. Solving Inequalities

What is Our Objective? We will use properties of equality, along with the solving steps of equations, in finding solutions to multiplication and division inequalities.

Previously…….. Our process for solving an inequality was the same as if the statement were an equation. The Equality Properties for inequalities were much like the Equality Properties for equations. Our process for solving an inequality was the same as if the statement were an equation. The Equality Properties for inequalities were much like the Equality Properties for equations. a = 4 b = 8 a < b a + c < a + b c = 3 4 < < < 11

This looks familiar…..notice the c values are (+) < 3 < 4 a = 4 b = 2 c = 2 If a > b a (c) > b (c) a = 6 b = 8 c = 2

? -4 ? -2 a = 4 b = 2 c = -1 If a > b a (c) > b (c) a = 4 b = 2 c = -1 However... When we use a (-)…. However... When we use a (-)…. < <

Inverting the Sign (Flip It!!) Test this. Let’s say a = 5 and b = 4 5 > 4 5(-1) ? 4(-1) -5 > is NOT greater than -4. Because we multiplied each term by a negative value, we invert the inequality sign. Test this. Let’s say a = 5 and b = 4 5 > 4 5(-1) ? 4(-1) -5 > is NOT greater than -4. Because we multiplied each term by a negative value, we invert the inequality sign. <

Furthermore..... Test this. Let’s say a = -8 and b = > -10 This is a true statement. > 4 < 5 Because we divided each term by a negative, we had to roll the sign over to the opposite sign. Test this. Let’s say a = -8 and b = > -10 This is a true statement. > 4 < 5 Because we divided each term by a negative, we had to roll the sign over to the opposite sign.

Keeping it True To keep the inequality accurate, when the SOLVING STEP of an inequality involves YOU Multiplicating or dividing by a negative number, the inequality sign must be flipped over as this solving step is completed! WHAT???? To keep the inequality accurate, when the SOLVING STEP of an inequality involves YOU Multiplicating or dividing by a negative number, the inequality sign must be flipped over as this solving step is completed! WHAT????

Try Some….. p. 112 t ≥ 7 -4 The solving step for this inequality WILL involve multiplication by a negative. (-4) t-28?≤ We will practice all the solving skills together. Relax. Have fun!!!

Remember….only in certain cases do you flip the sign!! Your solving step MUST involve mult/div by a negative!! m2 t7 ≥ <(-3) -3 4 m8 ≥t-21 The sign has to flip. You used a (-) to multiply. > (4)

5 < (7) 35 < r We will solve problems 1-6, and on p We will graph 4-6 and t > t > n < n > > -4n < n

4. 6m > n ≤ < -2m 6 m > 4 9 n ≤ > m 14. n > 3 -6 (-6) n < m ≤ (6) m ≤ n ≥5 3 n ≥ 15 (3)

Almost finished! y > 3 4 (4) y > r > 2 -4 (-4) r < > q -3 (-3) -18 < q

What Was Our Objective? We used properties of equality, along with the solving steps of equations, in finding solutions to multiplication and division inequalities. Remember! When you use a negative value in solving these inequalities, you must Flip the Sign!!