34 Lesson 1.8 ~ Linear Inequalities in One Variable

Nathan has more than $10 in his wallet. Jackie has run at most 200 miles this year. Each of these statements can be written using an inequality. Inequalities are mathematical statements which use >, $10 j ≤ 200Inequalities have multiple answers that can make the statement true. In Nathan's example, he might have $20 or $100. All that is known for certain is that he has more than $10 in his wallet. In Jackie's example, she might have run 200 miles this year or 5 miles. There are an infinite number of possibilities that make each statement true.

write an inequality for each statement.a. Carla's weight (w) is greater than 100 pounds.b. Vicky has at most $500 in her savings account. Let m represent the amount of money in Vicky's account.c. quinton's age is greater than 40 years old. Let a represent quinton's age.

a. The key words are “greater than”. Use the symbol >. w > 100

b. The key words are “at most”. This means she has less than or equal to $500. Use the ≤ symbol. m ≤ 500 c. The key words are “greater than”. Use the > symbol. a > 40

linEar inEqualiTiEs in onE variablE

Lesson 1.8

ExamplE 1

solutions

Lesson 1.8 ~ Linear Inequalities in One Variable 35

It is not possible to list all of the solutions to an inequality. In example 1c, Quinton could be 41, 42 or even 43.5 years old. All the answers can be shown on a number line.

a > 40

39 40 41 42 43 44 45 46

Forty is not included in the solution because Quinton's age is greater than 40, not equal to 40. This is shown on the number line with an "open circle" at 40. All of the numbers to the right of 40 are greater than 40 so they are included in the solution. This is shown with a line and an arrow pointing to the rest of the values.

When using the > or < inequality symbols, an "open circle" is used on the number line because the solution does not include the given number. When using the ≥ or ≤ inequality symbols, a "closed (or filled in) circle" is used because the solution contains the given number. Determining which direction the arrow should point is based on the relationship between the variable and the solution. The arrow points towards the set of numbers that make the statement true.

Inequalities are solved using properties similar to those you used to solve equations. Use inverse operations to isolate the variable so the solution can be graphed on a number line.

solve the inequality and graph its solution on a number line. x _ 4 + 2 ≥ 3 Subtract 2 from both sides of the inequality. x _ 4 + 2 ≥ 3 −2 −2 Multiply both sides of the inequality by 4. 4 ∙ x _ 4 ≥ 1 ∙ 4 x ≥ 4 Graph the solution on a number line. Use a closed circle.

−1 0 1 2 3 4 5 6

ExamplE 2

solution

36 Lesson 1.8 ~ Linear Inequalities in One Variable

solve the inequality and graph its solution on a number line. 6x + 3 < 2x − 5

Subtract 2x from each side of the inequality. 6x + 3 < 2x − 5 −2x −2x 4x + 3 < −5Subtract 3 from each side. −3 −3 4

4x < 8

4−

Divide both sides by 4. x < −2

Graph the solution on a number line. Use an open circle.

−5 −4 −3 −2 −1 0 1 2

One special rule applies to solving inequalities. Whenever you multiply or divide by a negative number on both sides of the equation, you must flip the inequality symbol. For example, less than () if you multiply or divide by a negative number.

solve the inequality −4x + 7 ≤ 19. Subtract 7 from each side of the inequality. −4x + 7 ≤ 19 −7 −7Divide both sides by −4. −4x ____ −4 ≤

12 ___ −4 Since both sides were divided by a negative,flip the inequality symbol. x ≥ −3

ExErcisEs

write an inequality for each graph shown. use x as the variable. 1.

−4 −3 −2 −1 0 1 2 3 2.

−3 −2 −1 0 1 2 3 4

3. −3 −2 −1 0 1 2 3 4

4. −3 −2 −1 0 1 2 3 4

ExamplE 3

solution

ExamplE 4

solution

Lesson 1.8 ~ Linear Inequalities in One Variable 37

solve each inequality. Graph the solution on a number line. 5. 4x − 1 ≥ 15 6. 10 < 6 + 2x 7. 3x + 10 < −2

8. 5x − 7 > 8 9. x __ 2 − 1 ≥ −1 10. −3x − 4 < 5 11. 3(x + 1) ≥ 9 12. 3 < 1 + x ___ −4 13. 5x < 2x − 21

14. −2 + 4x ≥ 3 − 6x 15. 2(x + 3) ≥ 5x + 12 16. 1 _ 2 x + 2 < x

17. A forklift has a maximum carrying capacity of 960 pounds. Each cargo box weighs 60 pounds. a. Write and solve an inequality that represents the maximum number of cargo boxes the forklift can hold. b. A 120-pound carrying case is used to hold the cargo boxes. What is the maximum number of cargo boxes the forklift can carry when the carrying case is used? Show that your answer is correct by showing that one more than your answer would exceed the forklift's capacity.

18. Olivia has $700 in her bank account at the beginning of the summer. She wants to have at least $150 in her account at the end of the summer. Each week she withdraws $40 for food and entertainment. a. Write an inequality for this situation. Let x represent the number of weeks she withdraws money from her account. b. What is the maximum number of weeks that Olivia can withdraw money from her account? Explain how you know your answer is correct. c. How much money will be left in her account after the last full withdrawal?

19. Ivan was at the beach. He wanted to spend $12 or less on a beach bike rental. The company he chose to rent from charged an initial fee of $5 and an additional $0.45 per mile he rode. a. Write an inequality for this situation. Let x represent the number of miles ridden. b. How many miles can Ivan ride without going over his spending limit? Write your answer as a whole number.

20. Mike went to the arcade. He spent less than or equal to $30; but spent more than $25. Create a number line that shows all the possible amounts that Mike may have spent at the arcade. Use words and/or numbers to show how you determined your answer.

rEviEw

solve each equation. describe the number of solutions (one, none or infinitely many). 21. 4x − 3 = x + 18 22. 4(x − 1) = 4x − 8 23. 5x − 3x + 2 = 2x + 2

24. 1 _ 2 (x + 4) = 1 _ 2 x + 2 25. 6(x − 3) = x − 3 26. 9 + 6x = 6x + 2 + 4

38 Lesson 1.8 ~ Linear Inequalities in One Variable

tic-tAc-toe ~ com poun d inequA l iti e s

Compound inequalities contain two inequality symbols. An example of a compound inequality is 4 < x < 9. This can be read “x is greater than four and less than 9.” A solution to a

compound inequality is a value that makes the statement true. In this case, any number between 4 and 9 makes the statement true. For example, 5.5 is a solution, 4 < 5.5 < 9, because it is greater than 4 but less than 9.

To graph a compound inequality, place open or closed circles (depending on the inequality) on each end value and connect the circles with a line in between.

0 5 10When solving compound inequalities, you must isolate the variable in the middle part of the inequality. In order to maintain the balance of the inequality, you must perform operations on ALL THREE parts of the inequality (left, middle and right)

example: Solve 2 ≤ x − 1 < 11. Graph the solution. 2 ≤ x − 1 < 11 Add 1 to all three parts of the inequality. +1 +1 +1 3 ≤ x < 12

0 5 10

solve each inequality below. Graph the solution on a number line. 1. 2 ≤ x + 2 ≤ 6 2. 6 < 2x < 16 3. −5 ≤ x − 2 < −3

4. 5 < 3x + 5 ≤ 17 5. 1 < x _ 3 − 1 < 2 6. 1

_ 2 ≤ 2x − 3 ≤ 1

tic-tAc-toe ~ one doe s not Be longCreate ten game cards. Each game card needs three equations on it that require two or more steps to solve. Two of the equations on a card should represent equations that have the same number of solutions (one, none or infinitely many). The other equation should have a different number of solutions. Participants try to locate the equation that does not fit. The cards can be used as a game or as a full class activity like a warm-up. Change the placement of the equations that do not belong so they are not always in the same spot on the cards.