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Worksheet: Multistep Inequalities

Q1:

Find all values of that satisfy . Write your answer as an interval.

  • A
  • B
  • C
  • D
  • E

Q2:

Solve the inequality 5 βˆ’ 1 2 π‘₯ β‰₯ 1 0 for π‘₯ .

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

Q3:

Solve the following inequality: 𝑛 5 + 1 < 6 .

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

Q4:

Given that π‘₯ ∈ β„€ , write the solution set of 2 π‘₯ βˆ’ 6 ≀ π‘₯ βˆ’ 1 .

  • A { 5 , 6 , 7 }
  • B { 5 , 6 , 7 , … }
  • C { 4 , 3 , 2 , … }
  • D { 5 , 4 , 3 , … }
  • E { 4 , 3 , 2 }

Q5:

Determine the solution set of given that .

  • A
  • B
  • C
  • D

Q6:

Suppose that π‘Ž > 𝑏 . Solve the inequality 𝑏 ( π‘₯ βˆ’ 5 ) β‰₯ π‘Ž π‘₯ + 3 𝑏 .

  • A π‘₯ β‰₯ 8 𝑏 𝑏 βˆ’ π‘Ž
  • B π‘₯ ≀ 8 𝑏 π‘Ž βˆ’ 𝑏
  • C π‘₯ ≀ βˆ’ 2 𝑏 𝑏 βˆ’ π‘Ž
  • D π‘₯ ≀ 8 𝑏 𝑏 βˆ’ π‘Ž
  • E π‘₯ ≀ 8 𝑏 𝑏 + π‘Ž

Q7:

The solution set of the inequality is , where and are real numbers. What is the value of ?

Q8:

Solve the inequality 1 0 π‘₯ + 1 6 ≀ 8 ( π‘₯ βˆ’ 1 9 ) in β„š .

  • A { π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ βˆ’ 6 8 }
  • B  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ βˆ’ 3 5 9 
  • C { π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ βˆ’ 8 4 }
  • D { π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ βˆ’ 8 4 }

Q9:

Solve the inequality 6 π‘₯ βˆ’ 2 7 4 β‰₯ 4 5 in β„š .

  • A  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ βˆ’ 1 5 1 3 0 
  • B  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ βˆ’ 1 1 9 3 0 
  • C  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ 1 5 1 3 0 
  • D  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ 1 5 1 3 0 
  • E  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ βˆ’ 1 1 9 3 0 

Q10:

If 𝑧 + 1 ≀ βˆ’ 9 , then βˆ’ 𝑧 .

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

Q11:

Find the solution set of the inequality in . Give your answer in interval notation.

  • A
  • B
  • C
  • D

Q12:

Solve the inequality 7 π‘₯ βˆ’ 8 π‘₯ + 1 1 ≀ 8 in β„š .

  • A  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ βˆ’ 1 5 
  • B { π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ βˆ’ 1 9 }
  • C  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ 1 5 
  • D { π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ 3 }
  • E { π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ βˆ’ 1 9 }

Q13:

Given that 𝑧 ∈ β„š , solve the inequality βˆ’ 4 ( 𝑧 βˆ’ 3 ) βˆ’ ( βˆ’ 4 𝑧 βˆ’ 4 ) ≀ βˆ’ 3 ( 3 𝑧 βˆ’ 1 ) .

  • A  𝑧 ∢ 𝑧 ∈ β„š , 𝑧 ≀ βˆ’ 2 3 
  • B  𝑧 ∢ 𝑧 ∈ β„š , 𝑧 β‰₯ βˆ’ 1 9 9 
  • C  𝑧 ∢ 𝑧 ∈ β„š , 𝑧 < βˆ’ 1 3 9 
  • D  𝑧 ∢ 𝑧 ∈ β„š , 𝑧 ≀ βˆ’ 1 3 9 
  • E  𝑧 ∢ 𝑧 ∈ β„š , 𝑧 β‰₯ βˆ’ 2 3 

Q14:

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

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

Q15:

Find the solution set of the inequality in . Give your answer in interval notation.

  • A
  • B
  • C
  • D

Q16:

Solve the inequality βˆ’ 1 0 ( π‘₯ + 2 ) < 1 6 π‘₯ βˆ’ 2 2 in β„š .

  • A  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ > 1 2 1 3 
  • B  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ < 1 1 3 
  • C  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ > 2 1 1 3 
  • D  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ > 1 1 3 

Q17:

Solve the inequality βˆ’ 4 ( 𝑦 + 4 ) βˆ’ 1 2 < βˆ’ 5 0 βˆ’ ( 4 7 βˆ’ 𝑦 ) in β„š .

  • A  𝑦 ∢ 𝑦 ∈ β„š , 𝑦 > βˆ’ 2 5 3 
  • B { 𝑦 ∢ 𝑦 ∈ β„š , 𝑦 > 2 3 }
  • C  𝑦 ∢ 𝑦 ∈ β„š , 𝑦 > 8 9 5 
  • D  𝑦 ∢ 𝑦 ∈ β„š , 𝑦 > 6 9 5 
  • E  𝑦 ∢ 𝑦 ∈ β„š , 𝑦 < 6 9 5 

Q18:

Solve the inequality 1 7 + 7 ( π‘₯ βˆ’ 1 3 ) β‰₯ π‘₯ + 4 4 in β„š .

  • A { π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ 8 }
  • B  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ 2 0 3 
  • C { π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ 5 }
  • D  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ 5 9 3 
  • E  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ 5 9 3 

Q19:

Solve the inequality 9 π‘₯ βˆ’ 3 ( βˆ’ 7 π‘₯ + 9 ) < βˆ’ 7 ( βˆ’ 9 + π‘₯ ) βˆ’ 2 in β„š .

  • A  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ > 1 0 4 3 7 
  • B  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ < 1 0 4 3 7 
  • C  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ > βˆ’ 3 4 5 
  • D  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ < 8 8 3 7 
  • E  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ < βˆ’ 3 4 5 

Q20:

Find the solution set of the inequality in . Give your answer in interval notation.

  • A
  • B
  • C
  • D
  • E

Q21:

Solve the inequality π‘₯ 8 βˆ’ 8 ≀ βˆ’ 7 π‘₯ βˆ’ 2 9 in β„š .

  • A  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ βˆ’ 2 9 6 5 7 
  • B  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ βˆ’ 2 9 6 5 7 
  • C  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ 1 6 8 5 5 
  • D  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ βˆ’ 5 6 1 9 
  • E  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ 1 6 8 5 5 

Q22:

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

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

Q23:

If 4 π‘₯ + 7 ≀ βˆ’ 1 , then 5 π‘₯ ≀ .

Q24:

Find the solution set of the inequality in . Give your answer in interval notation.

  • A
  • B
  • C
  • D

Q25:

Solve the inequality βˆ’ 6 ( π‘₯ βˆ’ 3 ) β‰₯ 4 ( π‘₯ + 5 ) in β„š .

  • A  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ 1 9 5 
  • B  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ βˆ’ 1 9 5 
  • C  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ βˆ’ 1 5 
  • D  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ ≀ βˆ’ 1 5 
  • E  π‘₯ ∢ π‘₯ ∈ β„š , π‘₯ β‰₯ βˆ’ 1 9 5 