Worksheet: Compound Linear Inequalities

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 βˆ’ 8 + π‘₯ < 7 π‘₯ + 4 ≀ 1 6 + π‘₯
  • B βˆ’ 8 + π‘₯ ≀ βˆ’ 7 π‘₯ + 4 ≀ 1 6 + π‘₯
  • C βˆ’ 8 + π‘₯ ≀ 7 π‘₯ + 4 < 1 6 + π‘₯
  • D βˆ’ 8 + π‘₯ ≀ 7 π‘₯ βˆ’ 4 ≀ 1 6 + π‘₯
  • E βˆ’ 8 + π‘₯ ≀ 7 π‘₯ + 4 ≀ 1 6 + π‘₯

Q2:

Solve βˆ’5βˆ’3π‘₯β‰₯4 and π‘₯+3<βˆ’5.

  • Ano solution
  • B π‘₯ β‰₯ βˆ’ 3
  • C π‘₯ ≀ βˆ’ 3
  • D π‘₯ < βˆ’ 8
  • E π‘₯ > βˆ’ 8

Q3:

Find the solution set of βˆ’6>2π‘₯+10>βˆ’16, where π‘₯βˆˆβ„•.

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

Q4:

Find the solution set of 5≀2π‘₯+3≀7 given that π‘₯βˆˆβ„•.

  • A { 2 }
  • B { 1 , 2 }
  • C βˆ…
  • D { 1 }

Q5:

Solve 10π‘₯+1<π‘₯βˆ’8 and π‘₯βˆ’1>βˆ’5.

  • A βˆ’ 4 > π‘₯ > βˆ’ 1
  • B π‘₯ < βˆ’ 1
  • C βˆ’ 4 < π‘₯ < βˆ’ 1
  • D π‘₯ > βˆ’ 4
  • E π‘₯ > βˆ’ 3

Q6:

Find the solution set of 7<π‘₯βˆ’12<11, where π‘₯βˆˆβ„•.

  • 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 π‘§βˆˆβ„š, solve the inequality 5β‰₯6𝑧+1β‰₯4.

  • A  𝑧 ∢ 𝑧 ∈ β„š , 2 3 β‰₯ 𝑧 β‰₯ 1 2 
  • B  𝑧 ∢ 𝑧 ∈ β„š , 1 > 𝑧 > 5 6 
  • C  𝑧 ∢ 𝑧 ∈ β„š , 2 3 > 𝑧 > 1 2 
  • D  𝑧 ∢ 𝑧 ∈ β„š , 5 6 β‰₯ 𝑧 β‰₯ 2 3 
  • E  𝑧 ∢ 𝑧 ∈ β„š , 1 β‰₯ 𝑧 β‰₯ 5 6 

Q8:

How many integer solutions does the inequality 12<π‘₯<34 have?

  • Ainfinite
  • B3
  • C2
  • D0

Q9:

Find the solution set of the inequality √64≀π‘₯βˆ’9β‰€βˆš81 in ℝ. Give your answer in interval notation.

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

Q10:

The solution set of π‘Žβ‰€π‘₯+4<𝑏 is [6,10). What are the values of π‘Ž and 𝑏?

  • A π‘Ž = 1 4 , 𝑏 = 1 0
  • B π‘Ž = 1 0 , 𝑏 = 1 4
  • C π‘Ž = 1 0 , 𝑏 = βˆ’ 6
  • D π‘Ž = 2 , 𝑏 = 6
  • E π‘Ž = βˆ’ 2 , 𝑏 = 1 4

Q11:

Solve βˆ’3π‘₯βˆ’5<4 and βˆ’π‘₯βˆ’1<βˆ’5.

  • A π‘₯ < 4
  • B π‘₯ > βˆ’ 3
  • C π‘₯ > 4
  • D π‘₯ < βˆ’ 3
  • E π‘₯ = βˆ’ 3

Q12:

Which of the following diagrams represents the solution set of the inequalities 3π‘₯βˆ’2<10 and π‘₯+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 < π‘₯ < 9
  • B π‘₯ > 9
  • C 3 < π‘₯ < 6
  • D 3 < π‘₯ < 9
  • E π‘₯ > 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 π‘Ž years. Write an inequality that describes the range of values of π‘Ž when a child must wear a seatbelt when sitting in the rear of the car.

  • A 8 < π‘Ž < 1 8
  • B π‘Ž < 1 8
  • C 8 ≀ π‘Ž < 1 8
  • D 8 < π‘Ž ≀ 1 8
  • E π‘Ž > 1 8

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

  • A 1 < π‘Ž < 8
  • B π‘Ž < 8
  • C 1 < π‘Ž ≀ 8
  • D π‘Ž > 8
  • E 1 ≀ π‘Ž < 8

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

  • A 𝑀 ≀ 2 0
  • B 𝑀 β‰₯ 4
  • C 𝑀 > 2 0
  • D 𝑀 β‰₯ 2 0
  • E 𝑀 ≀ 4

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

  • A 1 ≀ π‘Ž < 4
  • B 1 < π‘Ž < 4
  • C π‘Ž > 4
  • D π‘Ž < 4
  • E 1 < π‘Ž ≀ 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 𝑑dollars 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 π‘₯ < 2 2 . 1
  • B π‘₯ > 7 . 9
  • C π‘₯ < 1 5
  • D π‘₯ < 7 . 9
  • E π‘₯ > 2 2 . 1

Q17:

Express the following symbolically: π‘₯ is greater than3 and less than18.

  • A 3 < π‘₯ < 1 8
  • B 3 > π‘₯ > 1 8
  • C 3 ≀ π‘₯ β‰₯ 1 8
  • D 3 β‰₯ π‘₯ ≀ 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 π‘₯ that satisfy the inequality 15≀3π‘₯+3<21.

  • A 4 ≀ π‘₯ < 6
  • B 6 ≀ π‘₯ < 8
  • C 1 2 ≀ π‘₯ < 1 8
  • D 9 ≀ π‘₯ < 1 5
  • E 1 8 ≀ π‘₯ < 2 4

Q20:

Find all the values of π‘₯ that satisfy the compound inequality 7π‘₯βˆ’5>βˆ’126π‘₯+3β‰₯15.or

  • A π‘₯ > βˆ’ 1
  • B π‘₯ < βˆ’ 1
  • C π‘₯ β‰₯ 3
  • D π‘₯ > 1
  • E π‘₯ β‰₯ 2

Q21:

Find all the values of π‘₯ that satisfy 5π‘₯<15 or 7π‘₯+3β‰₯52.

  • A π‘₯ < 1 5 or π‘₯β‰₯7
  • B π‘₯ < 3 or π‘₯β‰₯49
  • C π‘₯ < 3 or π‘₯β‰₯7
  • D π‘₯ < 1 0 or π‘₯β‰₯7
  • E π‘₯ < 1 5 or π‘₯β‰₯49

Q22:

Given that βˆ’1β‰€βˆ’6π‘₯10βˆ’1≀5, find the greatest possible value of π‘₯+8.

Q23:

Given that π‘₯ is a whole number and 925<π‘₯25<1425, what are the possible values of π‘₯?

  • 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 βˆ’4≀π‘₯<0, where π‘₯βˆˆβ„•.

  • A { βˆ’ 3 , βˆ’ 2 , βˆ’ 1 }
  • B βˆ…
  • C { βˆ’ 3 , βˆ’ 2 , βˆ’ 1 , 0 }
  • D { βˆ’ 4 , βˆ’ 3 , βˆ’ 2 }

Q25:

Find the solution set of βˆ’4≀π‘₯βˆ’7β‰€βˆ’1, where π‘₯βˆˆβ„€.

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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.