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98 Chapter 14 Linear Inequalities
14.1 Lesson
An inequality is a mathematical sentence that compares expressions. It contains the symbols <, >,^ ≤ , or ≥. To write an inequality, look for the following phrases to determine where to place the inequality symbol.
Key Vocabulary inequality solution of an inequality solution set graph of an inequality
EXAMPLE 1 Writing an Inequality
A number w minus 3.5 is less than or equal to − 2. Write this sentence as an inequality.
A number w minus 3.5 is less than or equal to −2.
w − 3.5 ≤ − 2
An inequality is w − 3.5 ≤ −2.
Write the word sentence as an inequality.
1. A number b is fewer than 30.4. 2. Twice a number k is at least − (^) —^7 10
A solution of an inequality is a value that makes the inequality true. An inequality can have more than one solution. The set of all solutions of an inequality is called the solution set.
Exercises 6 – 9
Reading
The symbol ≥/ means
“is not greater than or equal to.”
Value of x x + 5 ≥ − 2 Is the inequality true?
− 1 ≥ − 2 ✓
yes
− 2 ≥ − 2 ✓
yes
− 3 ≥/ − 2 ✗^
no
Inequality Symbols Symbol < > ≤ ≥
Key Phrases
● is less than ● is fewer than
● is greater than ● is more than
● is less than or equal to ● is at most ● is no more than
● is greater than or equal to ● is at least ● is no less than
Section 14.1 Writing and Graphing Inequalities 99
EXAMPLE 2 Checking Solutions
Tell whether − 4 is a solution of the inequality. a. x + 8 <^ − 3 b. −4.5 x >^ − 21 x + 8 <^ − 3 Write the inequality. −4.5 x >^ − 21
− 4 + 8 <
− 3 Substitute −4 for x. −4.5(−4) >
4 </ − 3 ✗^ Simplify. 18 >^ − 21 ✓
4 is not less than −3. 18 is greater than −21.
So, −4 is not a solution So, −4 is a solution of the inequality. of the inequality.
Tell whether − 6 is a solution of the inequality.
3. c + 4 <^ − 1 4. 5 − m ≤ 10 5. 21 ÷ x ≥ −3.
The graph of an inequality shows all of the solutions of the inequality on a number line. An open circle is used when a number is not a solution. A closed circle is used when a number is a solution. An arrow to the left or right shows that the graph continues in that direction.
Exercises 11–
EXAMPLE 3 Graphing an Inequality
Graph y ≤ − 3.
Graph the inequality on a number line.
6. b >^ − 8 7. g ≤ 1.4 8. r <^ − (^1) — 2 9. v ≥ √
—
Exercises 17–
− 7 − 6 − 5 − 4 − 3 − 2 − 1 0 1 2 3 Test a number to the left of −3. y = −4 is a solution.
Test a number to the right of −3. y = 0 is not a solution.
Use a closed circle because −3 is a solution.
− 7 − 6 − 5 − 4 − 3 − 2 − 1 0 1 2 3
Shade the number line on the side where you found the solution.
Section 14.1 Writing and Graphing Inequalities 101
Solve the equation. Check your solution.
28. r − 12 = 3 29. 4.2 + p = 2.5 30. n − 3 π = 7 π 31. MULTIPLE CHOICE Which linear function relates y to x? A y = −0.5 x − 3 B y = 2 x + 3 C y = 0.5 x − 3 D y = 2 x − 3
Tell whether the given value is a solution of the inequality.
22. 3 p > 5 + p ; p = 4 23. — y 2
≥ y − 11; y = 18
24. VIDEO GAME RATINGS Each rating is matched with the inequality that represents the recommended ages of players. Your friend is old enough to play “E 10+” games. Is your friend old enough to play “T” games? Explain. 25. SCUBA DIVING Three requirements for a scuba diving training course are shown. a. Write and graph three inequalities that represent the requirements. b. You can swim 10 lengths of a 25-yard pool. Do you satisfy the swimming requirement of the course? Explain. 26. LUGGAGE On an airplane, the maximum sum of the length, width, and height of a carry-on bag is 45 inches. Find three different sets of dimensions that are reasonable for a carry-on bag. 27. A number m is less than another number n. The number n is less than or equal to a third number p. a. Write two inequalities representing these relationships. b. Describe the relationship between m and p. c. Can m be equal to p? Explain.
x − 1 0 1 2 y − 5 − 3 − 1 1
x q 33 x q 66 x q 1010 x q 1313 x q 1717
w hh
The ESRB rating icons are registered trademarks of the Entertainment Software Association.
102 Chapter 14 Linear Inequalities
14.2 Lesson
Addition Property of Inequality Words If you add the same number to each side of an inequality, the inequality remains true.
Numbers −3 <^2 Algebra x −^ 3 >^ −^10
- 4 + 4 + 3 + 3 1 <^6 x^ >^ −^7
Subtraction Property of Inequality Words If you subtract the same number from each side of an inequality, the inequality remains true.
Numbers −3 <^1 Algebra x +^ 7 >^ −^20 − 5 − 5 − 7 − 7 −8 <^ − 4 x >^ − 27 These properties are also true for ≤ and ≥.
Study Tip
You can solve inequalities the same way you solve equations. Use inverse operations to get the variable by itself.
Study Tip
To check a solution, you check some numbers that are solutions and some that are not.
EXAMPLE 1 Solving an Inequality Using Addition
Solve x − 6 ≥ − 10. Graph the solution.
x − 6 ≥ − 10 Write the inequality.
- 6 + 6 Add 6 to each side. x ≥ − 4 Simplify.
The solution is x ≥ −4.
− 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 0 1 2
x ≥ − 4
Check: x = −5 is not a solution. Check:^ x^ =^ 0 is a solution.
Solve the inequality. Graph the solution.
1. b − 2 >^ − 9 2. m − 3.8 ≤ 5 3. (^) —^1 4
(^) y − (^1) — 4
Undo the subtraction.
14.2 Exercises
9 +(-6)=3 3 +(-3)= 4 +(-9)= 9 +(-1)=
104 Chapter 14 Linear Inequalities
1. REASONING Is the inequality r − 5 ≤ 8 the same as 8 ≤ r − 5? Explain. 2. WHICH ONE DOESN’T BELONG? Which inequality does not belong with the other three? Explain your reasoning.
c + 7 — 2
3 — 2
c + 7 — 2
3 — 2
3 — 2 ≥ c + 7 — 2
c − (^3) — 2
7 — 2
Graph the inequality.
3. x ≥ − 6 4. p < 2 5. 4 ≤ w
Solve the inequality. Graph the solution.
6. y − 3 ≥ 7 7. t − 8 >^ − 4 8. n + 11 ≤ 20 9. a + 7 >^ − 1 10. 5 <^ v − 1 — 2
1 — 5
(^) d + —^4 5
2 — 3 ≤^ g^ −^
1 — 3
13. m + 7 — 4
11 — 4
14. 11.2 ≤ k + 9. 15. h − 1.7 <^ −3.2 16. 0 >^ s + π 17. 5 ≥ u − 4. 18. ERROR ANALYSIS Describe and correct the error in graphing the solution of the inequality. 19. PELICAN The maximum volume of a great white pelican’s bill is about 700 cubic inches. a. A pelican scoops up 100 cubic inches of water. Write and solve an inequality that represents the additional volume the bill can contain. b. A pelican’s stomach can contain about one-third the maximum amount that its bill can contain. Write an inequality that represents the volume of the pelican’s stomach.
5 ≥ x − 5
✗ 10 ≥ x^8 9 10 11 12
Section 14.2 Solving Inequalities Using Addition or Subtraction 105
Solve the equation.
28. 6 = 3 x 29. (^) — r 5
= 2 30. 4 c = 15 31. 8 = (^2) — 3
b
32. MULTIPLE CHOICE Which fraction is equivalent to 3.8?
A (^) —^5 19
B^19 — 5
C^12 — 15
D^12 — 5
Write and solve an inequality that represents the value of x****.
20. The perimeter is less 21. The base is greater 22. The perimeter is less than 16 feet. than the height. than or equal to 5 feet.
4 ft
4 ft (^) x
x + 2
10 m
x
12 in. 12 in.
10 in. 10 in.
23. REASONING The solution of w + c ≤ 8 is w ≤ 3. What is the value of c? 24. FENCE The hole for a fence post is 2 feet deep. The top of the fence post needs to be at least 4 feet above the ground. Write and solve an inequality that represents the required length of the fence post. 25. VIDEO GAME You need at least 12,000 points to advance to the next level of a video game. a. Write and solve an inequality that represents the number of points you need to advance. b. You fi nd a treasure chest that increases your score by 60%. How does this change the inequality? 26. POWER A circuit overloads at 1800 watts of electricity. A microwave that uses 1100 watts of electricity is plugged into the circuit. a. Write and solve an inequality that represents the additional number of watts you can plug in without overloading the circuit. b. In addition to the microwave, what two appliances in the table can you plug in without overloading the circuit? 27. The maximum surface area of the solid is 15 π square millimeters. Write and solve an inequality that represents the height of the cylinder.
Appliance Watts Clock radio 50 Blender 300 Hot plate 1200 Toaster 800
TIME LEFT: 1 min.
CURRENT SCORE: 4500
2 mm^ h
Sections 14.1– 14.2 Quiz 107
14.1– 14.2 Quiz
Write the word sentence as an inequality.
1. A number x plus 1 is less than −13. 2. A number t minus 1.6 is at most 9.
Tell whether the given value is a solution of the inequality.
3. 12 n <^ −2; n = − 1 4. y + 4 <^ −3; y = − 7
Graph the inequality on a number line.
5. x >^ − 10 6. y ≤ 3 — 5 7. w < 6.
Solve the inequality.
8. x − 2 < 4 9. g + 14 ≥ 30 10. h − 1 ≤ − 9 11. s + 3 >^ − 7 12. v − 3 — 4
< 0 13.^3 —
2
< (^) p + —^1 2
14. WATERCRAFT You must be at least 14 years old to operate a personal watercraft in Florida. Write an inequality that represents this situation. 15. REASONING The solution of x − a > 4 is x > 11. What is the value of a? 16. MP3 PLAYER Your MP3 player can store up to 8 gigabytes of media. You transfer 3.5 gigabytes of media to the MP3 player. Write and solve an inequality that represents the amount of memory available on the MP3 player. 17. LIFEGUARD Three requirements for a lifeguard training course are shown.
a. Write and graph three inequalities that represent the requirements. b. You can swim 350 feet. Do you satisfy the swimming requirement of the course? Explain.
Lifeguard Training Requirements Swim at least 100 yards Tread water for at least 5 minutes Swim 10 yards or more underwater without taking a breath
LIFEGUARDS NEEDED LIFEGUARDS NEEDED Take Our Training Course NOW!!!
108 Chapter 14 Linear Inequalities
14.3 Lesson
Remember
Multiplication and division are inverse operations.
EXAMPLE 1 Solving an Inequality Using Multiplication
Solve x — 8
> (^) − 5. Graph the solution.
x — 8
(^) − 5 Write the inequality.
x — 8
> 8 ⋅ (−5) Multiply each side by 8.
x >^ − 40 Simplify.
The solution is x >^ −40.
Solve the inequality. Graph the solution.
1. a ÷ 2 < 4 2. n — 7
w — 5
Undo the division.
− 90 − 80 − 70 − 60 − 50 − 40 − 30 − 20 − 10 0 10
x > − 40
Check: x = −80 is not a solution. Check: x = 0 is a solution.
Multiplication and Division Properties of Inequality (Case 1) Words If you multiply or divide each side of an inequality by the same positive number, the inequality remains true. Numbers −6 < 8^ 6 >^ −^8
2
> − —^8
2 −12 < 16^ 3 >^ −^4
Algebra x — 2
< (^) − 9 4 x > (^) − 12
2 ⋅ — x
2
< 2 ⋅ (−9) 4 — x
4
> − —^12
4
x <^ − 18 x >^ − 3
These properties are also true for ≤ and ≥.
110 Chapter 14 Linear Inequalities
EXAMPLE 3 Solving an Inequality Using Multiplication
Solve (^) — y − 3
> 2. Graph the solution.
y — − 3
2 Write the inequality.
− 3 ⋅ — y
− 3
< − 3 ⋅ 2 Multiply each side by^ −3. Reverse the
inequality symbol.
y <^ − 6 Simplify.
The solution is y <^ −6.
Undo the division.
EXAMPLE 4 Solving an Inequality Using Division
Solve − 7 y ≤ − 35. Graph the solution.
− 7 y ≤ − 35 Write the inequality.
− 7 y — − 7
− 35 — − 7 Divide each side by −7. Reverse the inequality symbol.
y ≥ 5 Simplify.
The solution is y ≥ 5.
− 1 0 1 2 3 4 5 6 7 8 9
y ≥ 5
Check: y = 0 is not a solution.^ Check:^ y^ =^ 6 is a solution.
Solve the inequality. Graph the solution.
7. p — − 4
< 7 8. — x − 5
1 — 10 z 10. − 9 m > 63
11. − 2 r ≥ − 22 12. −0.4 y ≥ − 12
Undo the multiplication.
Exercises 27–
− 9 − 8 − 7 − 6 − 5 − 4 − 3 − 2 − 1 0 1
y < − 6
Check: y = −9 is a solution. Check: y = 0 is not a solution.
Section 14.3 Solving Inequalities Using Multiplication or Division 111
14.3^ Exercises
9 +(-6)=3 3 +(-3)= 4 +(-9)= 9 +(-1)=
1. VOCABULARY Explain how to solve (^) — x 6
2. WRITING Explain how solving 2 x <^ −8 is different from solving − 2 x < 8. 3. OPEN-ENDED Write an inequality that is solved using the Division Property of Inequality where the inequality symbol needs to be reversed.
Use a table to solve the inequality.
4. 4 x < 4 5. − 2 x ≤ 2 6. − 5 x > 15 7. (^) — x − 3
≥ 1 8. (^) — x − 2
2
9. x — 4
≤ —^3
8
Solve the inequality. Graph the solution.
10. 3 n > 18 11. c — 4
≤ − 9 12. 1.2 m < 12
13. −14 >^ x ÷ 2 14. w — 5
≥ −2.6 15. 5 < 2.5 k
16. 4 x ≤ −^3 — 2 17. 2.6 y ≤ −10.4 18. 10.2 >^ — b 3. 19. ERROR ANALYSIS Describe and correct the error in solving the inequality.
Write the word sentence as an inequality. Then solve the inequality.
20. The quotient of a number and 3 is at most 4. 21. A number divided by 8 is less than −2. 22. Four times a number is at least −12. 23. The product of 5 and a number is greater than 20. 24. CAMERA You earn $9.50 per hour at your summer job. Write and solve an inequality that represents the number of hours you need to work in order to buy a digital camera that costs $247.
x — 2
x — 2
x >^ − 10
Section 14.3 Solving Inequalities Using Multiplication or Division 113
Solve the equation.
48. − 4 w + 5 = − 11 49. 4( x − 3) = 21
50. v — 6
m + 300 — 4
52. MULTIPLE CHOICE Which measure can have more than one value for a given data set? A mean B median C mode D range 41. TRIP You and three friends are planning a trip. You want to keep the cost below $80 per person. Write and solve an inequality that represents the total cost of the trip. 42. REASONING Explain why the direction of the inequality symbol must be reversed when multiplying or dividing by the same negative number. 43. PROJECT Choose two musical artists to research. a. Use the Internet or a magazine to complete the table. b. Find the average number of copies sold per month for each CD. c. Use the release date to write and solve an inequality that represents the minimum average number of copies sold per month for each CD. d. In how many months do you expect the number of copies of the second top selling CD to surpass the current number of copies of the top selling CD?
**Artist Name of CD Release Date Current Number of Copies Sold
2.**
Describe all numbers that satisfy both inequalities. Include a graph with your description.
44. 3 m >^ −12 and 2 m < 12 45. n — 2
≥ −3 and (^) — n − 4
46. 2 x ≥ −4 and 2 x ≥ 4 47. — m − 4
(^) −5 and m — 4
g
h. the table. month
s
114 Chapter 14 Linear Inequalities
14.4 Lesson
You can solve multi-step inequalities the same way you solve multi-step equations.
EXAMPLE 1 Solving Two-Step Inequalities
a. Solve 5 x − 4 ≥ 11. Graph the solution. 5 x − 4 ≥ 11 Write the inequality.
- 4 + 4 Add 4 to each side. 5 x ≥ 15 Simplify. 5 x — 5
≥ —^15
5
Divide each side by 5.
x ≥ 3 Simplify.
The solution is x ≥ 3.
− 3 − 2 − 1 0 1 2 3 4 5 6 7
x ≥ 3
Check: x = 0 is not a solution. Check: x = 4 is a solution.
b. Solve (^) — y − 6
- 7 < 9. Graph the solution.
y — − 6
- 7 <^9 Write the inequality.
− 7 − 7 Subtract 7 from each side. y — − 6
< 2 Simplify.
− 6 ⋅ — y
− 6
> − 6 ⋅ 2 Multiply each side by^ −6. Reverse the
inequality symbol.
y >^ − 12 Simplify.
The solution is y >^ −12.
− 18 − 16 − 14 − 12 − 10 − 8 − 6 − 4 − 2 0 2
y > − 12
Solve the inequality. Graph the solution.
1. 4 b − 1 < 7 2. 8 + 9 c ≥ − 28 3. (^) — n − 2
Step 1: Undo the subtraction.
Step 2: Undo the multiplication.
Exercises 5–
14.4 Exercises
9 +(-6)=3 3 +(-3)= 4 +(-9)= 9 +(-1)=
116 Chapter 14 Linear Inequalities
1. WRITING Compare and contrast solving multi-step inequalities and solving multi-step equations. 2. OPEN-ENDED Describe how to solve the inequality 3( a + 5) < 9. 3. For what values of k will the 4. For what values of h will the perimeter of the octagon be surface area of the solid be less than or equal to 64 units? greater than 46 square units? k
k
(^12) k (^12) k
4
4
4
4
5
3
h
Solve the inequality. Graph the solution.
5. 7 b + 4 ≥ 11 6. 2 v − 4 < 8 7. 1 − m — 3
8.^4 —
5
< 3 w − —^11 5
9. 1.8 < 0.5 − 1.3 p 10. −2.4 r + 9.6 ≥ 4. 11. ERROR ANALYSIS Describe and correct the error in solving the inequality.
Solve the inequality. Graph the solution.
12. 6( g + 2) ≤ 18 13. 2( y − 5) ≤ 16 14. − 10 ≥ 5 — 3
( h − 3)
15. − —^1
3
( u + 2) > 5 16. 2.7 > 0.9( n − 1.7) 17. 10 >^ −2.5( z − 3.1)
18. ATM Write and solve an inequality that represents the number of $20 bills you can withdraw from the account without going below the minimum balance.
x — 4
x + 6 ≥ 1 2 x ≥ 6
74 ft
8 ft
Section 14.4 Solving Multi-Step Inequalities 117
Find the area of the circle.
26.
10 mm
25 in.
66 m
29. MULTIPLE CHOICE What is the volume of the cube? A 8 ft^3 B 16 ft^3 C 24 ft^3 D 32 ft^3
Solve the inequality. Graph the solution.
19. 5 x − 2 x + 7 ≤ 15 + 10 20. 7 b − 12 b + 1.4 > 8.4 − 22 21. TYPING One line of text on a page uses about (^) —^3 16
of an inch. There are 1-inch margins at the top and bottom of a page. Write and solve an inequality to find the number of lines that can be typed on a page that is 11 inches long.
22. WOODWORKING A woodworker builds a cabinet in 20 hours. The cabinet is sold at a store for $500. Write and solve an inequality that represents the hourly wage the store can pay the woodworker and still make a profi t of at least $100. 23. FIRE TRUCK The height of one story of a building is about 10 feet. The bottom of the ladder on the fi re truck must be at least 24 feet away from the building. Write and solve an inequality to find the number of stories the ladder can reach. 24. DRIVE-IN A drive-in movie theater charges $3.50 per car. The drive-in has already admitted 100 cars. Write and solve an inequality to find the number of cars the drive-in needs to admit to make at least $500. 25. For what values of r will the area of the shaded region be greater than or equal to 9( π − 2)? r
2 ft
