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:

Jacob and Michael are competing on a quiz app. Jacob has 400 points and is losing 2 points per minute; Michael 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 Jacob has no fewer points than Michael.

  • 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 Michael catches up with Jacob. Assume that points are won or lost at a constant rate.

  • A6 minutes
  • B1212 minutes
  • C1512 minutes
  • D1213 minutes
  • E14 minutes

Q18:

Charlotte 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?

Q19:

Isabella has saved $31 in her piggy bank and her sister Scarlett has saved $36. If Scarlett saves $6 a week and Isabella saves $9 a week, after how many weeks will Isabella have saved more than Scarlett?

Q20:

Matthew wants to spend some of his birthday money to buy stationery. He buys a pencil case and some crayons to go in it. A pencil case costs $2 and crayons cost 75 cents each. Matthew has $10.

Write an inequality that can be used to find 𝑛, the number of crayons he can buy.

  • A10+0.75𝑛β‰₯2
  • B2+0.75𝑛≀10
  • C2+0.75𝑛<10
  • D10+0.5𝑛≀2
  • E10+0.5𝑛>2

Solve your inequality to find the maximum number of crayons that Matthew can buy.

Q21:

Daniel finds two plumbers online: the first charges $20 per hour of labor, while the second charges a fixed charge of $40 per job plus an hourly labor charge of $15. After how many hours will the second plumber be cheaper than the first?

Q22:

Noah and Jennifer were saving their allowances. Noah has started with $150 in his account and deposited $20 at the end of every month; Jennifer started with $50 in her account and deposited $32 at the end of every month.

Write an inequality that can be used to find π‘š, the number of months for which there was more money in Noah’s account than in Jennifer’s.

  • A150+20π‘š>50+32π‘š
  • B20βˆ’150π‘š>50+32π‘š
  • C75βˆ’20π‘šβ‰₯50+64π‘š
  • D150+20π‘š<50+32π‘š
  • E140+30π‘š<50+64π‘š

Use your inequality to find π‘š.

Q23:

When the sum of a number and twelve is multiplied by four, the result is more than when three times the number is subtracted from eleven.

Write an inequality to represent the statement above. Let π‘₯ represent the number.

  • A4(π‘₯+12)>3π‘₯βˆ’11
  • B4(π‘₯+12)<3π‘₯βˆ’11
  • C4(π‘₯+12)≀7+5π‘₯
  • D4(π‘₯+12)>11βˆ’3π‘₯
  • E4(π‘₯+12)β‰₯11βˆ’3π‘₯

Q24:

In a board game, Mason scored 22, 11, 23, 19, and 17 points in five turns. Find the minimum number of points he must score in the sixth turn to have an average of at least 17 points.

Q25:

Anthony has a total of $200 and wants to buy some Blu-ray discs. Given that Blu-ray discs cost $18.75 each, and he must save at least $65, write an inequality that can be used to find how many Blu-ray discs he can buy, and then determine the maximum number of Blu-ray discs he can buy.

  • A18.75π‘₯βˆ’200≀65, 7 Blu-ray discs
  • B200βˆ’18.75π‘₯β‰₯65, 7 Blu-ray discs
  • C200βˆ’18.75π‘₯≀65, 7 Blu-ray discs
  • D18.75π‘₯βˆ’200≀65, 14 Blu-ray discs
  • E18.75π‘₯βˆ’200β‰₯65, 14 Blu-ray discs

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