Worksheet: Two-Step Linear Inequalities

In this worksheet, we will practice solving a linear inequality in two steps.

Q1:

Find the solution set of 3π‘₯βˆ’7<βˆ’4 given that π‘₯βˆˆβ„•.

  • A { 0 }
  • B { βˆ’ 1 , 0 }
  • C { 1 }
  • D { 0 , 1 }

Q2:

Determine the solution set of 2βˆ’π‘₯β‰€βˆ’8, where π‘₯βˆˆβ„€οŠ°.

  • A { 1 1 , 1 2 , 1 3 , … }
  • B { 1 0 , 1 1 , 1 2 , … }
  • C { 9 , 8 , 7 , … }
  • D βˆ…

Q3:

Find the solution set of βˆ’5π‘₯βˆ’4>1 given that π‘₯βˆˆβ„•.

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

Q4:

Mrs. Williams tells her math class, β€œFive more than four times a number is more than 12.” Let π‘₯ represent the number and write an inequality that represents her statement.

  • A 4 π‘₯ + 5 β‰₯ 1 2
  • B 4 π‘₯ + 5 ≀ 1 2
  • C 3 π‘₯ + 5 > 2 4
  • D 4 π‘₯ + 5 > 1 2
  • E 5 π‘₯ + 4 β‰₯ 1 2

Q5:

Given that π‘₯βˆˆβ„š, solve the inequality 7π‘₯βˆ’5≀8.

  • A  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ 3 7 
  • B  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ 1 3 7 
  • C  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ < 1 3 7 
  • D  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ < 3 7 
  • E  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ 8 7 

Q6:

Solve the inequality 19+7π‘₯<40 in β„š.

  • A { π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ > 3 }
  • B  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ < 5 9 7 
  • C { π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ < 3 }
  • D { π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ > βˆ’ 3 }

Q7:

Find the solution set of the inequality 13π‘₯+1<βˆ’6 in ℝ. Give your answer in interval notation.

  • A ( βˆ’ ∞ , βˆ’ 2 1 )
  • B ( βˆ’ ∞ , βˆ’ 2 1 ]
  • C [ βˆ’ 2 1 , ∞ )
  • D ( βˆ’ 2 1 , ∞ )

Q8:

Find the solution set of the inequality 2π‘₯βˆ’2>4 in ℝ. Give your answer in interval notation.

  • A [ 3 , ∞ )
  • B ( 3 , ∞ )
  • C [ 3 , ∞ ]
  • D ( 1 , ∞ )
  • E ( 3 , ∞ ]

Q9:

Solve 7≀2π‘₯+1.

  • A βˆ’ 3 ≀ π‘₯
  • B π‘₯ ≀ βˆ’ 3
  • C 3 ≀ π‘₯
  • D π‘₯ ≀ 3

Q10:

Solve the inequality 5βˆ’12π‘₯β‰₯10 for π‘₯.

  • A π‘₯ ≀ 1 0
  • B π‘₯ ≀ 1 0 3
  • C π‘₯ β‰₯ βˆ’ 1 0
  • D π‘₯ ≀ βˆ’ 1 0
  • E π‘₯ ≀ βˆ’ 1 5

Q11:

Solve the following inequality: 𝑛5+1<6.

  • A 𝑛 > 1 2
  • B 𝑛 > 2 5
  • C 𝑛 < 3 5
  • D 𝑛 < 2 5
  • E 𝑛 > 3 5

Q12:

Write the set of all integer solutions to 4π‘₯βˆ’8<βˆ’4.

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

Q13:

Solve 5π‘₯+7<βˆ’3. Write your answer as an interval.

  • A { βˆ’ 2 }
  • B ( βˆ’ 2 , ∞ )
  • C ( βˆ’ ∞ , βˆ’ 2 )
  • D ( βˆ’ ∞ , βˆ’ 2 ]
  • E [ βˆ’ 2 , ∞ )

Q14:

Determine the solution set of 9π‘₯+5β‰₯βˆ’4 given that π‘₯βˆˆβ„€.

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

Q15:

Find the solution set of the inequality βˆ’2π‘₯+3≀5. Write your answer as an interval.

  • A [ βˆ’ 1 , ∞ )
  • B ( βˆ’ 1 , ∞ )
  • C { βˆ’ 1 }
  • D ( βˆ’ ∞ , βˆ’ 1 )
  • E ( βˆ’ ∞ , βˆ’ 1 ]

Q16:

Find the solution set of 2π‘₯βˆ’1β‰€βˆ’9 given that π‘₯βˆˆβ„•.

  • A { βˆ’ 6 , βˆ’ 5 }
  • B βˆ…
  • C { 4 }
  • D { 6 , 5 }

Q17:

If 4π‘₯+7β‰€βˆ’1, then 5π‘₯≀.

Q18:

If 𝑧+1β‰€βˆ’9, then βˆ’π‘§.

  • A β‰₯ 1 0
  • B β‰₯ βˆ’ 1 0
  • C ≀ 1 0
  • D ≀ βˆ’ 1 0

Q19:

Determine the solution set of βˆ’7π‘₯+5>βˆ’9 given that π‘₯βˆˆβ„€.

  • A { 0 , 1 , 2 }
  • B { 1 , 0 , βˆ’ 1 , … }
  • C { βˆ’ 1 , 0 , 1 }
  • D { 2 , 1 , 0 , … }
  • E { 3 , 4 , 5 , … }

Q20:

Michael wants to purchase a precision balance for $340. He has already saved $113 and can save $28 every week. Write an inequality that can be used to determine the number of weeks left for Michael to save at least $340.

  • A 1 1 3 π‘₯ + 2 8 β‰₯ 3 4 0
  • B 2 8 π‘₯ + 1 1 3 β‰₯ 3 4 0
  • C 2 8 π‘₯ + 1 1 3 > 3 4 0
  • D 1 1 3 π‘₯ + 2 8 ≀ 3 4 0
  • E 2 8 π‘₯ + 1 1 3 ≀ 3 4 0

Q21:

Matthew needs to buy some clothes. The store’s parking lot has the shown sign outside.

Write an inequality for 𝑑, the time in hours, that Matthew can park if he has only $8.25 in cash.

  • A 1 . 5 ( 𝑑 βˆ’ 1 ) ≀ 8 . 2 5
  • B 1 . 5 ( 𝑑 βˆ’ 1 ) β‰₯ 8 . 2 5
  • C 1 . 5 ( 𝑑 + 1 ) < 8 . 2 5
  • D ( 𝑑 + 1 ) > 8 . 2 5
  • E ( 𝑑 βˆ’ 1 ) ≀ 8 . 2 5

Given that you must pay for whole hours of parking, use your inequality to find the maximum time that Matthew can park.

  • A6 hours
  • B5 hours
  • C12 hours
  • D9 hours
  • E7 hours

Q22:

A hotel caters for large parties and events. They charge $300 for the hall and $15 per person for a lunchtime buffet.

Write an inequality that can be used to find 𝑛, the number of people who can go to a party that was planned with a budget of $1,000.

  • A 3 0 0 + 1 5 𝑛 ≀ 1 , 0 0 0
  • B 1 5 + 1 , 0 0 0 𝑛 ≀ 3 0 0
  • C 1 5 + 1 , 0 0 0 𝑛 β‰₯ 3 0 0
  • D 3 0 0 + 1 5 𝑛 β‰₯ 1 , 0 0 0
  • E 1 5 + 3 0 0 𝑛 ≀ 1 , 0 0 0

Use your inequality to find the maximum number of people.

Q23:

A candy store has a special offer: if you spend more than $15, you get a free chocolate drink. Gift boxes are $3 each, and chocolates are $2 per 50 g. Write an inequality to find 𝑀, the weight of the chocolate you must buy with a gift box, if you are to receive a free chocolate drink.

  • A 1 0 0 𝑀 + 3 > 1 5
  • B 𝑀 2 5 + 3 > 1 5
  • C 5 0 𝑀 + 3 > 1 5
  • D 𝑀 2 5 + 3 β‰₯ 1 5
  • E 2 𝑀 + 3 β‰₯ 1 5

Q24:

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

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

Q25:

Solve the following inequality: 9𝑦>77+2𝑦.

  • A 𝑦 < 1 1
  • B 𝑦 > 7
  • C 𝑦 < 7
  • D 𝑦 > 7 7
  • E 𝑦 > 1 1

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