Worksheet: Multistep Inequalities

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In this worksheet, we will practice solving multistep inequalities.

Q1:

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

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

Q2:

Solve the inequality 5π‘šβˆ’9(π‘š+3)<14 in β„š.

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

Q3:

Find the solution set of the inequality βˆ’14π‘₯βˆ’52β‰€βˆ’18π‘₯ in ℝ. Give your answer in interval notation.

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

Q4:

Solve the inequality 17+7(π‘₯βˆ’13)β‰₯π‘₯+44 in β„š.

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

Q5:

Solve the inequality 7π‘₯βˆ’8π‘₯+11≀8 in β„š.

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

Q6:

Solve the inequality 10π‘₯+16≀8(π‘₯βˆ’19) in β„š.

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

Q7:

Solve the inequality 6π‘₯βˆ’274β‰₯45 in β„š.

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

Q8:

Given that π‘§βˆˆβ„š, solve the inequality βˆ’4(π‘§βˆ’3)βˆ’(βˆ’4π‘§βˆ’4)β‰€βˆ’3(3π‘§βˆ’1).

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

Q9:

Solve the inequality βˆ’10(π‘₯+2)<16π‘₯βˆ’22 in β„š.

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

Q10:

Solve the inequality βˆ’4(𝑦+4)βˆ’12<βˆ’50βˆ’(47βˆ’π‘¦) in β„š.

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

Q11:

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

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

Q12:

Solve the inequality π‘₯8βˆ’8β‰€βˆ’7π‘₯βˆ’29 in β„š.

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

Q13:

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

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

Q14:

Find the solution set of the inequality βˆ’3π‘₯+11β‰€βˆ’π‘₯+37 in ℝ. Give your answer in interval notation.

  • A ( 2 4 , ∞ )
  • B [ βˆ’ 2 4 , ∞ )
  • C [ βˆ’ 1 3 , ∞ )
  • D ( βˆ’ ∞ , βˆ’ 1 3 ]
  • E ( βˆ’ 1 3 , ∞ )

Q15:

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

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

Q16:

Given that π‘¦βˆˆβ„š, solve the inequality βˆ’3π‘¦βˆ’9<7π‘¦βˆ’4.

  • A 𝑦 ≀ βˆ’ 1 3 4
  • B 𝑦 < βˆ’ 1 3 4
  • C 𝑦 > βˆ’ 1 2
  • D 𝑦 < βˆ’ 1 2
  • E 𝑦 > 2 5

Q17:

When hired at a new job selling electronics, you are given two pay options:

  • Option A: a base salary of $10,000 a year with a commission of 9% of your sales.
  • Option B: a base salary of $20,000 a year with a commission of 4% of your sales.

How many dollars’ worth of electronics would you need to sell for Option A to produce a larger income?

  • AMore than $76,923.08 worth of electronics
  • BMore than $230,769.23 worth of electronics
  • CMore than $200,000 worth of electronics
  • DMore than $600,000 worth of electronics

Q18:

When hired at a new job selling electronics, you are given two pay options:

  • Option A: a base salary of $20,000 a year with a commission of 12% of your sales
  • Option B: a base salary of $26,000 a year with a commission of 3% of your sales

How many dollars’ worth of electronics would you need to sell for option A to produce a larger income?

  • Amore than $306,666.67 worth of electronics
  • Bmore than $511,111.11 worth of electronics
  • Cmore than $40,000 worth of electronics
  • Dmore than $66,666.67 worth of electronics

Q19:

A cell phone company offers the following two plans:

  • Plan A: $15 per month and $2 for every 300 texts.
  • Plan B: $25 per month and $0.50 for every 100 texts.

How many texts would you need to send per month for plan B to save you money?

  • Aless than 60
  • Bmore than 6,000
  • Cmore than 60
  • Dless than 6,000

Q20:

William and Daniel are competing on a quiz app. William has 400 points and is losing 2 points per minute; Daniel has 250 points and is winning 10 points per minute.

Write an inequality which can be used to find π‘š, the amount of time for which William has no fewer points than Daniel.

  • A 4 0 0 + 2 π‘š < 2 5 0 + 1 0 π‘š
  • B 4 0 0 βˆ’ 2 π‘š β‰₯ 2 5 0 + 1 0 π‘š
  • C 2 0 0 βˆ’ 2 π‘š β‰₯ 2 5 0 + 1 5 π‘š
  • D 4 0 0 + 3 π‘š > 2 5 0 + 1 0 π‘š
  • E 2 0 0 βˆ’ 3 π‘š ≀ 2 5 0 + 1 5 π‘š

Use your inequality to find the time when Daniel catches up with William. Assume that points are won or lost at a constant rate.

  • A6 minutes
  • B 1 2 1 2 minutes
  • C 1 5 1 2 minutes
  • D 1 2 1 3 minutes
  • E14 minutes

Q21:

A cell phone company offers the following two plans:

  • Plan A: $20 per month and $1 for every one hundred texts.
  • Plan B: $50 per month with free, unlimited texts.

How many texts would you need to send per month for plan B to save you money?

  • AMore than 3,000
  • BLess than 3,000
  • CLess than 30
  • DMore than 30

Q22:

Sophia finds two landscape gardeners online: the first charges a fixed fee of $20 per job plus $15 per hour for labor, while the second charges a fixed fee of $90 but only $5 per hour for labor. After how many hours will the second gardener be cheaper than the first?

Q23:

You just got a new job in retail and you are given two options for your earnings:

Option 1: Base salary of $15,000 per year plus a commission of 14% of your sales.

Option 2: Base salary of $18,000 per year plus a commission of 10% of your sales.

How much money in sales would you need to make in order for Option 1 to yield a higher income than Option 2?

  • AMore than $137,500
  • BMore than $12,500
  • CMore than $75,000
  • DMore than $18,000

Q24:

A moving company charges a flat rate of $150 and an additional $5 for each box. If a taxi service charged $20, for how many boxes would it be cheaper to move with the taxi service than with the moving company?

  • Amore than 6 boxes
  • Bonly at 10 boxes
  • Cmore than 10 boxes
  • Dless than 10 boxes
  • Eless than 6 boxes

Q25:

The cost of producing β€œdoomflots” is $325 for equipment plus another $32 for materials and labor for each one produced. How many β€œdoomflots” need to be produced such that the average cost per β€œdoomflot” is less than $50?

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