Worksheet: Rational Equations

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In this worksheet, we will practice solving rational equations.

Q1:

Find the solution set of the equation 2π‘₯+2+5π‘₯βˆ’5=35π‘₯βˆ’3π‘₯βˆ’10.

  • Aβˆ…
  • B{5}
  • C{2}
  • D{βˆ’4}
  • E{5,βˆ’5}

Q2:

Given that 7π‘₯π‘₯βˆ’3=16π‘₯π‘₯+3βˆ’9, find the value of π‘₯.

  • A2717
  • B2711
  • C3
  • D275
  • E2723

Q3:

What is the solution set of the equation 1π‘₯βˆ’1βˆ’1π‘₯=1π‘₯βˆ’1π‘₯+1?

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

Q4:

Solve the equation 12=23π‘₯βˆ’1.

  • Aπ‘₯=94
  • Bπ‘₯=19
  • Cπ‘₯=49
  • Dπ‘₯=14
  • Eπ‘₯=35

Q5:

Find the value of π‘₯ that makes 0.2π‘₯βˆ’6π‘₯+4 equal to zero.

Q6:

Solve 2π‘₯+13=1.

  • Aπ‘₯=3
  • Bπ‘₯=1
  • Cπ‘₯=23
  • Dπ‘₯=0
  • Eπ‘₯=13

Q7:

Solve the equation 8π‘šβˆ’1+4π‘š+1=βˆ’8π‘šβˆ’1.

  • Aπ‘š=0
  • Bπ‘š=1
  • Cπ‘š=βˆ’2
  • Dπ‘š=βˆ’1
  • Eno solution

Q8:

Solve 3π‘›βˆ’4+1𝑛+2=2π‘›βˆ’2.

  • A𝑛=βˆ’2
  • B𝑛=3
  • Cno solution
  • D𝑛=βˆ’3
  • E𝑛=2

Q9:

Solve 3π‘₯+1+6π‘₯βˆ’1=2 for π‘₯.

  • Aπ‘₯=βˆ’13, π‘₯=1
  • Bπ‘₯=βˆ’12, π‘₯=5
  • Cπ‘₯=15, π‘₯=5
  • Dπ‘₯=12, π‘₯=3
  • Eπ‘₯=βˆ’12, π‘₯=5

Q10:

If 2π‘₯+75π‘₯βˆ’9=12, find the value of π‘₯.

Q11:

Find the solution set of 1(π‘₯βˆ’12)=0.0001οŠͺ in ℝ.

  • A{2,22}
  • B{βˆ’22,βˆ’2}
  • C{βˆ’2}
  • D{22}

Q12:

Find π‘₯ given π‘₯βˆ’20π‘₯+10=0.

Q13:

Solve 2π‘₯+13π‘₯+4=35.

  • Aπ‘₯=βˆ’239
  • Bπ‘₯=βˆ’179
  • Cπ‘₯=1
  • Dπ‘₯=7
  • Eπ‘₯=17

Q14:

Solve 125π‘₯+3=5.

  • Aπ‘₯=22
  • Bπ‘₯=628
  • Cπ‘₯=622
  • Dπ‘₯=βˆ’22
  • Eπ‘₯=25

Q15:

Find the value of π‘₯ given π‘₯+44π‘₯ is the multiplicative inverse of 913.

  • A3643
  • B94
  • C5223
  • Dβˆ’5223

Q16:

Liam went on a bike ride of 48 miles. He realized that if he had gone 4 mph faster, then he would have arrived 6 hours sooner. How fast did he actually ride?

Q17:

If Victoria does a job in 10 hours, and with the help of Scarlett, they can do it together in 6 hours, how long would it take Scarlett to do it alone?

Q18:

It takes Victoria  2 hours to pick up 8 kg of strawberries. When she is helped by her son, Ethan, it takes them 1 hour and a quarter. How long would it take Ethan to pick up 8 kg of strawberries alone? Give your answer in hours and minutes.

  • A3 hours and 20 minutes
  • B2 hours and 45 minutes
  • C2 hours and 30 minutes
  • D3 hours and 45 minutes
  • E3 hours and 15 minutes

Q19:

A speech therapist charges two different fees, $20 and $35, depending on the type of the session. On a given day, she has already had 3 sessions at $20 and 4 at $35. For the rest of the day, she will only have sessions at $20. How many of those should she give to earn an average rate of $25 per session?

Q20:

Consider the equation 1π‘₯βˆ’3βˆ’2π‘₯+3=3+π‘₯9βˆ’π‘₯.

If you multiply both sides by π‘₯βˆ’9 and rearrange so that the right-hand side is 0, what equation do you get?

  • A12=0
  • Bπ‘₯+3=0
  • Cπ‘₯βˆ’3=0
  • Dπ‘₯βˆ’π‘₯βˆ’6=0
  • E0=0

What can you say about the solutions of the original equation?

  • Athere are two solutions
  • Bthere are no solutions
  • Cevery value of π‘₯ is a solution
  • Dthere is only one solution

Q21:

For what value of π‘Ÿ does the equation 2π‘₯+5π‘₯βˆ’3=π‘Ÿ have no solutions?

Q22:

Find the solution set of 2π‘₯βˆ’2+20π‘₯+2=6 in ℝ.

  • A3,23
  • B{βˆ’6,βˆ’2}
  • Cο¬βˆ’3,βˆ’23
  • Dο¬βˆ’3,23
  • E{6,2}

Q23:

Find the solution set of π‘₯+93π‘₯+38=12 in ℝ.

  • A{4,βˆ’5}
  • B4,βˆ’52
  • Cο¬βˆ’4,βˆ’52
  • Dο¬βˆ’4,52
  • E{βˆ’4,5}

Q24:

Given that the multiplicative inverse of π‘₯βˆ’π‘π‘₯βˆ’4 is π‘₯βˆ’4π‘₯βˆ’2, what is the value of 𝑐?

Q25:

Given that π‘šπ‘› is a rational number and π‘šπ‘›=116, find all the possible values of ο€»π‘šπ‘›ο‡οŠ©.

  • A14
  • B164
  • C164,βˆ’164
  • D14,096,βˆ’14,096

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