Lesson 10: Solving Inequalities
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1 Opening Exercise 1. Find the solution set to each inequality. Express the solution graphically on the number line and give the solution in interval notation. A. xx B. mm > 9 C. 8yy + 4 < 7yy 2 D. 6(xx 5) 30 E. 4(xx 3) > 2(xx 2) Unit 3: Solving Equations & Inequalities S.81
2 2. Stephanie says, So far we have the following rules for inequalities: If AA > BB, then AA + cc > BB + cc for any real number cc. If AA > BB, then kkaa > kkbb for any positive real number k. Explain to your partner what Stephanie means by these statements. Be prepared to share out with the class. Stephanie is quite clear in her rules that you cannot multiply by a negative number. Let s see what happens if we do multiply by a -1 with the inequality 5 C > 2. multiplied by -1 gives us the inequality 5 CC > CC > 2 3. Find one number that works for the first one. Does it work for the second one? 5 CC > CC > 2 4. A. Let s look at why this happens. If we choose two numbers on the number line, let s say 2 and 4 and mark them. We can see that 2 < 4. B. Now multiply our two numbers by -1 and mark these new numbers on the number line. You should now see that -4 < -2. What does multiplying by 1 do to our inequality? Unit 3: Solving Equations & Inequalities S.82
3 The Properties of Inequalities Addition property of inequality: If AA > BB, then AA + cc > BB + cc for any real number cc. Multiplication property of inequality: If AA > BB, then kkaa > kkbb for any positive real number kk. 5. Use the properties of inequality to show that each of the following is true for any real numbers pp and qq. A. If pp qq, then pp qq. B. If pp < qq, then 5pp > 5qq. 6. Based on the results from Exercises 4 and 5, how might we expand the multiplication property of inequality? Multiplication property of inequality: If AA > BB, then kkaa > kkbb for any positive real number kk. Multiplication property of inequality: If AA > BB, then for any negative real number kk. 7. Find the solution set to each inequality. Express the solution graphically on a number line. A. 2ff < 16 B. xx C. 6 aa 15 D. 3(2xx + 4) > 0 Unit 3: Solving Equations & Inequalities S.83
4 Lesson Summary The Properties of Inequalities Addition property of inequality: If AA > BB, then AA + cc > BB + cc for any real number cc. Multiplication property of inequality: If AA > BB, then kkaa > kkbb for any positive real number kk. If AA > BB, then kkaa < kkbb for any negative real number kk. Unit 3: Solving Equations & Inequalities S.84
5 Homework Problem Set Find the solution set to each inequality. Express the solution graphically on the number line and give the solution in interval notation. 1. 2xx < xx xx (xx 1) xx < 9(1 xx) Unit 3: Solving Equations & Inequalities S.85
6 6. Solve xx + 1 5xx, for xx without multiplying by a negative number. Then, solve by multiplying on 16 2 both sides by Lisa brought half of her savings to the bakery and bought 12 croissants for $ The amount of money she brings home with her is more than $2.00. Use an inequality to find how much money she had in her savings before going to the bakery. (Write the inequality that represents the situation, and solve it.) 8. Solve 18 16t > 6 100t, for tt in two different ways: first without ever multiplying on both sides by a negative number and then by first multiplying on both sides by 1 or dividing by 2. 2 Unit 3: Solving Equations & Inequalities S.86
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