Multi-step inequalities

Variables on both sides

Compound Inequalities

Absolute-value inequalities CHALLENGE

10 10 10 10 10

20 20 20 20 20

30 30 30 30 30

40 40 40 40 40

50 50 50 50 50

Solve and graph solutions

-12 ≥ 3x + 6

Answer:

x ≤ -6

-6-7 -5

Write the Inequality shown by the graph

-5 0 5

Answer

x < -5

Solve and graph the inequality

-¼(p + 10) ≥ 6 – 4

Answer

p ≤ -18

-18 -17-19

Question

Notebooks cost $1.39 each. What are the possible numbers of notebooks that can be purchased with $10?

Answer

n = notebooksn ≤ 7

Up to 7 notebooks can be purchased with $10.

Question

Carl’s Cable Company charges $55 for monthly service plus $4 for each pay-per-view movie. Teleview Cable Company charges $110 per month with no fee for movies. For what number of movies is the cost of Carl’s Cable Company less than the cost of Teleview.

Answer

m = moviesm < 13.75

You could rent up to 13 movies for Carl’s Cable Company to be less than Teleview.

Solve each inequality and graph the solutions.

-3(2 – q) ≥ 6(q - 1)

Answer

0 ≥ q

0-1 1

Solve and graph the inequality

5(4 + k) < 5k

Answer

No Solutions

Solve the inequality and graph the solutions.

3.5t – 1.8 < 1.6t + 3.9

Answer

t < 3

32 4

Question

Hanna has a savings account with a balance of $210 and deposits $16 per month. Faith has a savings account with a balance of $175 and deposits $20 per month. Write and solve an inequality to determine the number of months Hanna’s account balance will be greater than Faith’s account balance.

Answer

m = monthsm < 8.75

Hannah’s account will be greater than Faith’s until the 9th month.

Question

In an acute triangle, all angles measure less than 90°. Also, the sum of the measures of any two angles is greater than the measure of the third angle. Can the measures of an acute triangle be x, x-1, and 2x.

Answer

No, because the sum of the angles x and x-1 is not greater than 2x.

Solve the compound inequality and graph the solutions.

7x ≥ 21 OR 2x < -2

Answer

x ≥ 3 OR x < -1

3-1

Solve the compound inequality and graph the solutions

5 < 3x – 1 < 17

Answer

2 < x < 6

62

Decide whether the three lengths given can form a triangle. If not,

explain.

6½ yd, 3 yd, 2¾ yd

Answer

No, because the sides with measure 3 yd and 2¾ yd is not greater than the length of the 3rd side.

Question

The cruise-control function on Georgina’s car should keep the speed of the car within 3 mi/h of the set speed. Write a compound inequality to show the acceptable speeds s if the set speed is 55 mi/h.

Answer

-3 < s – 55 < 3

Question

The ball used in a soccer game may not weigh more than 16 ounces or less than 14 ounces at the start of the match. After 1½ ounces of air was added to a ball, the ball was approved for use in a game. Write and solve a compound inequality to show how much the ball might have weighed before the air was added.

Answer

14 ≤ x + 1½ ≤ 1612½ ≤ x ≤ 14½

The ball might have weighed anywhere from 12½ ounces to 14½ ounces.

Solve and graph the solutions

2 I x I ≤ 6

Answer

-3 ≤ x ≤ 3

3-3

Solve and graph the solutions

4 + I x + 3 I > 7

Answer

x < -6 OR x > 0

0-6

Solve and graph the solutions

4 I x – 3.5 I ≤ -8

Answer

The inequality has no solutions.

Question

A nutritionist recommends that an adult male consumes 55 grams of fat per day. It is acceptable for the fat intake to differ from this amount by at most 25 grams. Write and solve an absolute-value inequality to find the range of fat intake that is acceptable.

Answer

I x – 55 I ≤ 2530 ≤ x ≤ 80

It is acceptable for an adult male to intake anywhere from 30 to 80 grams of fat per day.

QuestionThe frequency of a sound wave determines its pitch. The human ear can detect a wide range of frequencies, from 20 Hz (very low notes) to 20,000 Hz (very high notes)

a.) What frequency is at the middle of the rangeb.) Write an absolute-value inequality for the range of frequencies the human ear can detect.

Answer

a.) 10100b.) I x – 10010 I ≤ 9990

Solve and graph the inequality

- 18 > - (2x + 9) – 4 + x

Answer

x > 5

54 6

Question

Replace the square and circle with number so that the inequality has

all real numbers as solutions.☐ - 2x < - 2x

Answer

Answers may vary:

☐ = -1 = 0

Question

For the compound inequality x + 2 ≥ a AND x – 7 ≤ b, find values of a and b for which the only solution is x = 1.

Answer

a = 3b = -6

Question

Write a compound inequality that represents all values of x that are NOT solutions to x < -1 OR x > 3.

Answer

-1 ≤ x ≤ 3

Question

The water depth for a pool is set to 6 ft, but the actual depth of the pool may vary by as much as 4 in. Write and solve an absolute-value inequality to find the range of possible water depths in inches.

Answer

I x – 72 I ≤ 468 ≤ x ≤ 76

The actual depth of the pool could be anywhere from 68 inches to 76 inches.