Lesson 14: Solving Inequalities

Transcription

1 Hart Interactive Algebra 1 Lesson 14 Classwork 1. Consider the inequality xx 2 + 4xx 5. a. Think about some possible values to assign to xx that make this inequality a true statement. Find at least two positive values that work and at least two negative values that work. b. Should your four values also be solutions to the inequality 4xx + xx 2 5? Explain why or why not. Are they? Commutative property c. Should your four values also be solutions to the inequality xx(xx + 4) 5? Explain why or why not. Are they? Yes. Distributive property d. Should your four values also be solutions to the inequality 4xx + xx 2 6 1? Explain why or why not. Are they? Yes. Addition property e. Should your four values also be solutions to the inequality 12xx + 3xx 2 15? Explain why or why not. Are they? Multiplied by What is the solution set to the inequality 5qq + 10 > 20? Express the solution set in words, in set notation, and graphically on the number line. Words: the solution to this inequality are any values that are greater than 2. Set notation: { q } Graphic: S.115

2 > < open circle; Closed circle 3. Find the solution set to each inequality. Express the solution in set notation and graphically on the number line. a. xx b. mm c. 8yy + 4 < 7yy 2 d. 6(xx 5) 30 e. 4(xx 3) > 2(xx 2) 4. 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. When adding a same number to both sides of an inequality, we are not changing the solution set. When multiplying a positive number to both sides of an inequality, the solution set is not changed. Stephanie is quite clear in her rules that you cannot multiplying 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 S.116

3 5. Find one number that works for the first one. Does it work for the second one? 5 CC > CC > 2 6. A. Let s look at why this happens. Use the line below to create a number line with numbers from -5 to 5. 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. 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. 7. 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. S.117

4 8. Based on the results from Exercises 6 and 7, 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. 9. Find the solution set to each inequality. Express the solution in set notation and graphically on the number line. ***when x or. by a negative number, flip inequality sign a. 2ff < 16 b. xx {f R f>8}. {x R x. -3} c. 6 aa 15 d. 3(2xx + 4) > Jojo was asked to solve 6xx + 12 < 3xx + 6, for xx. She answered as follows: 6xx + 12 < 3xx + 6 6(xx + 2) < 3(xx + 2) Apply the distributive property. 6 < 3 Multiply through by 1 xx+2. a. Since the final line is a false statement, she deduced that there is no solution to this inequality (that the solution set is empty). What is the solution set to 6xx + 12 < 3xx + 6? b. Explain why Jojo came to an erroneous conclusion. S.118

5 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. Homework Problem Set 1. Find the solution set to each inequality. Express the solution in set notation and graphically on the number line. a. 2xx < 10 b. 15xx 45 c. 2 3 xx d. 5(xx 1) 10 e. 13xx < 9(1 xx) S.119

6 2. Find the mistake in the following set of steps in a student s attempt to solve 5xx + 2 xx + 2, for xx. What 5 is the correct solution set? 5xx + 2 xx xx xx (factoring out 5 on the left side) 5 1 (dividing by xx ) So, the solution set is the empty set. 3. 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.) 5. Solve 4 + 2tt 14 18tt > 6 100tt, 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 2. S.120