Worksheet: Compound Linear Inequalities

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


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

  • Aβˆ’8+π‘₯<7π‘₯+4≀16+π‘₯
  • Bβˆ’8+π‘₯β‰€βˆ’7π‘₯+4≀16+π‘₯
  • Cβˆ’8+π‘₯≀7π‘₯+4<16+π‘₯
  • Dβˆ’8+π‘₯≀7π‘₯βˆ’4≀16+π‘₯
  • Eβˆ’8+π‘₯≀7π‘₯+4≀16+π‘₯


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

  • Ano solution
  • Bπ‘₯β‰₯βˆ’3
  • Cπ‘₯β‰€βˆ’3
  • Dπ‘₯<βˆ’8
  • Eπ‘₯>βˆ’8


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

  • A{βˆ’25,βˆ’24,…,βˆ’17}
  • B{βˆ’13,βˆ’12,…,βˆ’8}
  • C{βˆ’26,βˆ’25,…,βˆ’16}
  • Dβˆ…


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

  • A{2}
  • B{1,2}
  • Cβˆ…
  • D{1}


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

  • Aβˆ’4>π‘₯>βˆ’1
  • Bπ‘₯<βˆ’1
  • Cβˆ’4<π‘₯<βˆ’1
  • Dπ‘₯>βˆ’4
  • Eπ‘₯>βˆ’3


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

  • A{19,20,21,22,23}
  • B{19,20,21,22}
  • C{20,21,22,23}
  • D{20,21,22}


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

  • Aο¬π‘§βˆΆπ‘§βˆˆβ„š,23β‰₯𝑧β‰₯12
  • Bο¬π‘§βˆΆπ‘§βˆˆβ„š,1>𝑧>56
  • Cο¬π‘§βˆΆπ‘§βˆˆβ„š,23>𝑧>12
  • Dο¬π‘§βˆΆπ‘§βˆˆβ„š,56β‰₯𝑧β‰₯23
  • Eο¬π‘§βˆΆπ‘§βˆˆβ„š,1β‰₯𝑧β‰₯56


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


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(13,18]
  • C[13,18]
  • D[βˆ’5,0)


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

  • Aπ‘Ž=14, 𝑏=10
  • Bπ‘Ž=10, 𝑏=14
  • Cπ‘Ž=10, 𝑏=βˆ’6
  • Dπ‘Ž=2, 𝑏=6
  • Eπ‘Ž=βˆ’2, 𝑏=14


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

  • Aπ‘₯<4
  • Bπ‘₯>βˆ’3
  • Cπ‘₯>4
  • Dπ‘₯<βˆ’3
  • Eπ‘₯=βˆ’3


Which of the following diagrams represents the solution set of the inequalities 3π‘₯βˆ’2<10 and π‘₯+2>3?

  • A
  • B
  • C
  • D


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.

  • A6<π‘₯<9
  • Bπ‘₯>9
  • C3<π‘₯<6
  • D3<π‘₯<9
  • Eπ‘₯>6


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.

  • A8<π‘Ž<18
  • Bπ‘Ž>18
  • C8<π‘Žβ‰€18
  • D8β‰€π‘Ž<18
  • Eπ‘Ž<18

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.

  • A1<π‘Žβ‰€8
  • Bπ‘Ž<8
  • Cπ‘Ž>8
  • D1<π‘Ž<8
  • E1β‰€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.

  • A𝑀≀4
  • B𝑀>20
  • C𝑀β‰₯20
  • D𝑀≀20
  • E𝑀β‰₯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.

  • A1<π‘Ž<4
  • B1β‰€π‘Ž<4
  • Cπ‘Ž<4
  • Dπ‘Ž>4
  • E1<π‘Žβ‰€4


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.

  • A17≀𝑇≀7
  • B0≀𝑇≀17
  • C7≀𝑇≀17
  • D7≀𝑇≀30
  • E0≀𝑇≀30


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.

  • Aπ‘₯<22.1
  • Bπ‘₯>7.9
  • Cπ‘₯<15
  • Dπ‘₯<7.9
  • Eπ‘₯>22.1


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

  • A3<π‘₯<18
  • B3>π‘₯>18
  • C3≀π‘₯β‰₯18
  • D3β‰₯π‘₯≀18


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?

  • A0.75≀𝑑<6
  • B0.5≀𝑑<3
  • C0.25<𝑑≀3
  • D0.25≀𝑑<3
  • E0.75<𝑑≀3


Find all the values of π‘₯ that satisfy the inequality 15≀3π‘₯+3<21.

  • A6≀π‘₯<8
  • B18≀π‘₯<24
  • C9≀π‘₯<15
  • D12≀π‘₯<18
  • E4≀π‘₯<6


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

  • Aπ‘₯>βˆ’1
  • Bπ‘₯<βˆ’1
  • Cπ‘₯β‰₯3
  • Dπ‘₯>1
  • Eπ‘₯β‰₯2


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

  • Aπ‘₯<15 or π‘₯β‰₯7
  • Bπ‘₯<3 or π‘₯β‰₯49
  • Cπ‘₯<3 or π‘₯β‰₯7
  • Dπ‘₯<10 or π‘₯β‰₯7
  • Eπ‘₯<15 or π‘₯β‰₯49


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


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

  • A10,12,14,13
  • B11,12,13,14
  • C9,10,11,12
  • D10,11,12,13


Find the solution set of βˆ’4≀π‘₯<0, where π‘₯βˆˆβ„•.

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


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

  • A{3,4,5,6}
  • B{βˆ’11,βˆ’10,βˆ’9,βˆ’8}
  • C{4,5}
  • D{βˆ’11,βˆ’10,…,6}

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