Worksheet: Multistep Inequalities

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

Q2:

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

  • Aο¬π‘šβˆΆπ‘šβˆˆβ„š,π‘š>βˆ’4114
  • Bο¬π‘šβˆΆπ‘šβˆˆβ„š,π‘š>βˆ’1314
  • Cο¬π‘šβˆΆπ‘šβˆˆβ„š,π‘š<βˆ’1314
  • Dο¬π‘šβˆΆπ‘šβˆˆβ„š,π‘š>134
  • Eο¬π‘šβˆΆπ‘šβˆˆβ„š,π‘š>βˆ’414

Q3:

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

  • A[13,∞)
  • B(βˆ’βˆž,βˆ’13]
  • C(βˆ’βˆž,13)
  • D(βˆ’βˆž,13]

Q4:

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

  • Aπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯203
  • Bπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯593
  • C{π‘₯∢π‘₯βˆˆβ„š,π‘₯≀8}
  • D{π‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯5}
  • Eπ‘₯∢π‘₯βˆˆβ„š,π‘₯≀593

Q5:

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

  • A{π‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯βˆ’19}
  • B{π‘₯∢π‘₯βˆˆβ„š,π‘₯β‰€βˆ’19}
  • C{π‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯3}
  • Dπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯15
  • Eπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰€βˆ’15

Q6:

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

  • A{π‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯βˆ’84}
  • B{π‘₯∢π‘₯βˆˆβ„š,π‘₯β‰€βˆ’68}
  • C{π‘₯∢π‘₯βˆˆβ„š,π‘₯β‰€βˆ’84}
  • Dπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰€βˆ’359

Q7:

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

  • Aπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰€βˆ’11930
  • Bπ‘₯∢π‘₯βˆˆβ„š,π‘₯≀15130
  • Cπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯βˆ’11930
  • Dπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰€βˆ’15130
  • Eπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯15130

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).

  • Aο¬π‘§βˆΆπ‘§βˆˆβ„š,𝑧β‰₯βˆ’23
  • Bο¬π‘§βˆΆπ‘§βˆˆβ„š,π‘§β‰€βˆ’139
  • Cο¬π‘§βˆΆπ‘§βˆˆβ„š,π‘§β‰€βˆ’23
  • Dο¬π‘§βˆΆπ‘§βˆˆβ„š,𝑧<βˆ’139
  • Eο¬π‘§βˆΆπ‘§βˆˆβ„š,𝑧β‰₯βˆ’199

Q9:

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

  • Aπ‘₯∢π‘₯βˆˆβ„š,π‘₯>2113
  • Bπ‘₯∢π‘₯βˆˆβ„š,π‘₯>1213
  • Cπ‘₯∢π‘₯βˆˆβ„š,π‘₯>113
  • Dπ‘₯∢π‘₯βˆˆβ„š,π‘₯<113

Q10:

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

  • A{π‘¦βˆΆπ‘¦βˆˆβ„š,𝑦>23}
  • Bο¬π‘¦βˆΆπ‘¦βˆˆβ„š,𝑦>βˆ’253
  • Cο¬π‘¦βˆΆπ‘¦βˆˆβ„š,𝑦<695
  • Dο¬π‘¦βˆΆπ‘¦βˆˆβ„š,𝑦>895
  • Eο¬π‘¦βˆΆπ‘¦βˆˆβ„š,𝑦>695

Q11:

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

  • Aπ‘₯∢π‘₯βˆˆβ„š,π‘₯>βˆ’345
  • Bπ‘₯∢π‘₯βˆˆβ„š,π‘₯<10437
  • Cπ‘₯∢π‘₯βˆˆβ„š,π‘₯<8837
  • Dπ‘₯∢π‘₯βˆˆβ„š,π‘₯>10437
  • Eπ‘₯∢π‘₯βˆˆβ„š,π‘₯<βˆ’345

Q12:

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

  • Aπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰€βˆ’5619
  • Bπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯16855
  • Cπ‘₯∢π‘₯βˆˆβ„š,π‘₯≀16855
  • Dπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯βˆ’29657
  • Eπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰€βˆ’29657

Q13:

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

  • Aπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰€βˆ’195
  • Bπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯βˆ’195
  • Cπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰₯βˆ’15
  • Dπ‘₯∢π‘₯βˆˆβ„š,π‘₯β‰€βˆ’15
  • Eπ‘₯∢π‘₯βˆˆβ„š,π‘₯≀195

Q14:

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

  • A(24,∞)
  • B[βˆ’24,∞)
  • C[βˆ’13,∞)
  • D(βˆ’βˆž,βˆ’13]
  • E(βˆ’13,∞)

Q15:

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

  • Aπ‘₯≀8𝑏𝑏+π‘Ž
  • Bπ‘₯≀8π‘π‘Žβˆ’π‘
  • Cπ‘₯β‰₯8π‘π‘βˆ’π‘Ž
  • Dπ‘₯≀8π‘π‘βˆ’π‘Ž
  • Eπ‘₯β‰€βˆ’2π‘π‘βˆ’π‘Ž

Q16:

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

  • Aπ‘¦β‰€βˆ’134
  • B𝑦<βˆ’134
  • C𝑦>βˆ’12
  • D𝑦<βˆ’12
  • E𝑦>25

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.

  • A400+2π‘š<250+10π‘š
  • B400βˆ’2π‘šβ‰₯250+10π‘š
  • C200βˆ’2π‘šβ‰₯250+15π‘š
  • D400+3π‘š>250+10π‘š
  • E200βˆ’3π‘šβ‰€250+15π‘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
  • B1212 minutes
  • C1512 minutes
  • D1213 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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