Worksheet: Compound Linear Inequalities

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In this worksheet, we will practice solving compound linear inequalities by applying inverse operations.

Q1:

This number line describes the solution set to which of the following inequalities?

  • A − 8 + 𝑥 < 7 𝑥 + 4 ≤ 1 6 + 𝑥
  • B − 8 + 𝑥 ≤ − 7 𝑥 + 4 ≤ 1 6 + 𝑥
  • C − 8 + 𝑥 ≤ 7 𝑥 + 4 < 1 6 + 𝑥
  • D − 8 + 𝑥 ≤ 7 𝑥 − 4 ≤ 1 6 + 𝑥
  • E − 8 + 𝑥 ≤ 7 𝑥 + 4 ≤ 1 6 + 𝑥

Q2:

Solve −5−3𝑥≥4 and 𝑥+3<−5.

  • Ano solution
  • B 𝑥 ≥ − 3
  • C 𝑥 ≤ − 3
  • D 𝑥 < − 8
  • E 𝑥 > − 8

Q3:

Find the solution set of −6>2𝑥+10>−16, where 𝑥∈ℕ.

  • A { − 2 5 , − 2 4 , … , − 1 7 }
  • B { − 1 3 , − 1 2 , … , − 8 }
  • C { − 2 6 , − 2 5 , … , − 1 6 }
  • D ∅

Q4:

Find the solution set of 5≤2𝑥+3≤7 given that 𝑥∈ℕ.

  • A { 2 }
  • B { 1 , 2 }
  • C ∅
  • D { 1 }

Q5:

Solve 10𝑥+1<𝑥−8 and 𝑥−1>−5.

  • A − 4 > 𝑥 > − 1
  • B 𝑥 < − 1
  • C − 4 < 𝑥 < − 1
  • D 𝑥 > − 4
  • E 𝑥 > − 3

Q6:

Find the solution set of 7<𝑥−12<11, where 𝑥∈ℕ.

  • A { 1 9 , 2 0 , 2 1 , 2 2 , 2 3 }
  • B { 1 9 , 2 0 , 2 1 , 2 2 }
  • C { 2 0 , 2 1 , 2 2 , 2 3 }
  • D { 2 0 , 2 1 , 2 2 }

Q7:

Given that 𝑧∈ℚ, solve the inequality 5≥6𝑧+1≥4.

  • A 𝑧 ∶ 𝑧 ∈ ℚ , 2 3 ≥ 𝑧 ≥ 1 2
  • B 𝑧 ∶ 𝑧 ∈ ℚ , 1 > 𝑧 > 5 6
  • C 𝑧 ∶ 𝑧 ∈ ℚ , 2 3 > 𝑧 > 1 2
  • D 𝑧 ∶ 𝑧 ∈ ℚ , 5 6 ≥ 𝑧 ≥ 2 3
  • E 𝑧 ∶ 𝑧 ∈ ℚ , 1 ≥ 𝑧 ≥ 5 6

Q8:

How many integer solutions does the inequality 12<𝑥<34 have?

  • Ainfinite
  • B3
  • C2
  • D0

Q9:

Find the solution set of the inequality √64≤𝑥−9≤√81 in ℝ. Give your answer in interval notation.

  • A [ − 5 , 0 ]
  • B ( 1 3 , 1 8 ]
  • C [ 1 3 , 1 8 ]
  • D [ − 5 , 0 )

Q10:

The solution set of 𝑎≤𝑥+4<𝑏 is [6,10). What are the values of 𝑎 and 𝑏?

  • A 𝑎 = 1 4 , 𝑏 = 1 0
  • B 𝑎 = 1 0 , 𝑏 = 1 4
  • C 𝑎 = 1 0 , 𝑏 = − 6
  • D 𝑎 = 2 , 𝑏 = 6
  • E 𝑎 = − 2 , 𝑏 = 1 4

Q11:

Solve −3𝑥−5<4 and −𝑥−1<−5.

  • A 𝑥 < 4
  • B 𝑥 > − 3
  • C 𝑥 > 4
  • D 𝑥 < − 3
  • E 𝑥 = − 3

Q12:

Which of the following diagrams represents the solution set of the inequalities 3𝑥−2<10 and 𝑥+2>3?

  • A
  • B
  • C
  • D

Q13:

Emma has 6 cats as pets. One of her cats, Natalie, is pregnant. Write an inequality for the number of cats Emma will have if Natalie gives birth to fewer than 3 kittens.

  • A 6 < 𝑥 < 9
  • B 𝑥 > 9
  • C 3 < 𝑥 < 6
  • D 3 < 𝑥 < 9
  • E 𝑥 > 6

Q14:

A state’s regulations to ensure child passenger safety are as follows:

  • From the age of 18, there is no legal requirement for passengers traveling in the back seat of a car to use a seat belt.
  • Children under 1 year of age and children weighing less than 20 lb are required to be in a rear-facing seat.
  • Children age 1 to age 3 years (inclusive) and at least 20 lb in weight are required to be in a forward-facing seat.
  • Children age 4 to age 7 years (inclusive) are required to be in a forward-facing car seat or a booster seat.
  • Children at 8 years old or above are required to use standard vehicle safety belt.

The age of a child is 𝑎 years. Write an inequality that describes the range of values of 𝑎 when a child must wear a seatbelt when sitting in the rear of the car.

  • A 8 < 𝑎 < 1 8
  • B 𝑎 < 1 8
  • C 8 ≤ 𝑎 < 1 8
  • D 8 < 𝑎 ≤ 1 8
  • E 𝑎 > 1 8

The age of a child who weighs over 20 lb is 𝑎 years. Write an inequality that describes the range of values of 𝑎 when the child must use a forward-facing car seat or booster seat.

  • A 1 < 𝑎 < 8
  • B 𝑎 < 8
  • C 1 < 𝑎 ≤ 8
  • D 𝑎 > 8
  • E 1 ≤ 𝑎 < 8

A child is under 4 years old. The child’s weight in pounds is 𝑤. Write an inequality that describes the range of values of 𝑤, for which the child may travel in a forward-facing seat.

  • A 𝑤 ≤ 2 0
  • B 𝑤 ≥ 4
  • C 𝑤 > 2 0
  • D 𝑤 ≥ 2 0
  • E 𝑤 ≤ 4

A child weighs more than 20 pounds. Their age is 𝑎. Write an inequality that describes the range of ages for which they must use a forward-facing seat.

  • A 1 ≤ 𝑎 < 4
  • B 1 < 𝑎 < 4
  • C 𝑎 > 4
  • D 𝑎 < 4
  • E 1 < 𝑎 ≤ 4

Q15:

A warehouse stores two types of food. The first needs to be stored at a temperature between 0 and 17 degrees. As for the second, it needs to be kept at a temperature between 7 to 30 degrees. Find the range of the temperature 𝑇 at which the two types of food can be stored together.

  • A 1 7 ≤ 𝑇 ≤ 7
  • B 0 ≤ 𝑇 ≤ 1 7
  • C 7 ≤ 𝑇 ≤ 1 7
  • D 7 ≤ 𝑇 ≤ 3 0
  • E 0 ≤ 𝑇 ≤ 3 0

Q16:

Natalie had $15 when she went to the cinema. She spent 𝑡dollars on the ticket and $7.10 on the popcorn. Write an inequality that represents the amount of money Natalie had when she left the cinema.

  • A 𝑥 < 2 2 . 1
  • B 𝑥 > 7 . 9
  • C 𝑥 < 1 5
  • D 𝑥 < 7 . 9
  • E 𝑥 > 2 2 . 1

Q17:

Express the following symbolically: 𝑥 is greater than3 and less than18.

  • A 3 < 𝑥 < 1 8
  • B 3 > 𝑥 > 1 8
  • C 3 ≤ 𝑥 ≥ 1 8
  • D 3 ≥ 𝑥 ≤ 1 8

Q18:

Students in a grade 4 class were asked the distances in miles, 𝑑, which they travel to get to school. All of the students traveled farther than a quarter of a mile, and no one traveled farther than 3 miles. Which of the following inequalities represents the range of distances traveled to school?

  • A 0 . 7 5 ≤ 𝑑 < 6
  • B 0 . 5 ≤ 𝑑 < 3
  • C 0 . 2 5 < 𝑑 ≤ 3
  • D 0 . 2 5 ≤ 𝑑 < 3
  • E 0 . 7 5 < 𝑑 ≤ 3

Q19:

Find all the values of 𝑥 that satisfy the inequality 15≤3𝑥+3<21.

  • A 4 ≤ 𝑥 < 6
  • B 6 ≤ 𝑥 < 8
  • C 1 2 ≤ 𝑥 < 1 8
  • D 9 ≤ 𝑥 < 1 5
  • E 1 8 ≤ 𝑥 < 2 4

Q20:

Find all the values of 𝑥 that satisfy the compound inequality 7𝑥−5>−126𝑥+3≥15.or

  • A 𝑥 > − 1
  • B 𝑥 < − 1
  • C 𝑥 ≥ 3
  • D 𝑥 > 1
  • E 𝑥 ≥ 2

Q21:

Find all the values of 𝑥 that satisfy 5𝑥<15 or 7𝑥+3≥52.

  • A 𝑥 < 1 5 or 𝑥≥7
  • B 𝑥 < 3 or 𝑥≥49
  • C 𝑥 < 3 or 𝑥≥7
  • D 𝑥 < 1 0 or 𝑥≥7
  • E 𝑥 < 1 5 or 𝑥≥49

Q22:

Given that −1≤−6𝑥10−1≤5, find the greatest possible value of 𝑥+8.

Q23:

Given that 𝑥 is a whole number and 925<𝑥25<1425, what are the possible values of 𝑥?

  • A 1 0 , 1 2 , 1 4 , 1 3
  • B 1 1 , 1 2 , 1 3 , 1 4
  • C 9 , 1 0 , 1 1 , 1 2
  • D 1 0 , 1 1 , 1 2 , 1 3

Q24:

Find the solution set of −4≤𝑥<0, where 𝑥∈ℕ.

  • A { − 3 , − 2 , − 1 }
  • B ∅
  • C { − 3 , − 2 , − 1 , 0 }
  • D { − 4 , − 3 , − 2 }

Q25:

Find the solution set of −4≤𝑥−7≤−1, where 𝑥∈ℤ.

  • A { 3 , 4 , 5 , 6 }
  • B { − 1 1 , − 1 0 , − 9 , − 8 }
  • C { 4 , 5 }
  • D { − 1 1 , − 1 0 , … , 6 }

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