Worksheet: Rational Equations

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.

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

Q2:

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

  • A3
  • B 2 7 2 3
  • C 2 7 5
  • D 2 7 1 1
  • E 2 7 1 7

Q3:

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

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

Q4:

Solve the equation 12=23π‘₯βˆ’1.

  • A π‘₯ = 3 5
  • B π‘₯ = 1 4
  • C π‘₯ = 1 9
  • D π‘₯ = 4 9
  • E π‘₯ = 9 4

Q5:

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

Q6:

Solve 2π‘₯+13=1.

  • A π‘₯ = 1
  • B π‘₯ = 1 3
  • C π‘₯ = 2 3
  • D π‘₯ = 0
  • E π‘₯ = 3

Q7:

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

  • A π‘š = βˆ’ 2
  • B π‘š = βˆ’ 1
  • C π‘š = 0
  • D π‘š = 1
  • Eno solution

Q8:

Solve 3π‘›βˆ’4+1𝑛+2=2π‘›βˆ’2.

  • A 𝑛 = 3
  • B 𝑛 = βˆ’ 3
  • C 𝑛 = 2
  • D no solution
  • E 𝑛 = βˆ’ 2

Q9:

Solve 3π‘₯+1+6π‘₯βˆ’1=2 for π‘₯.

  • A π‘₯ = 1 2 , π‘₯ = 3
  • B π‘₯ = 1 5 , π‘₯ = 5
  • C π‘₯ = βˆ’ 1 2 , π‘₯ = 5
  • D π‘₯ = βˆ’ 1 2 , π‘₯ = 5
  • E π‘₯ = βˆ’ 1 3 , π‘₯ = 1

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

Q12:

Find π‘₯ given π‘₯βˆ’20π‘₯+10=0.

Q13:

Solve 2π‘₯+13π‘₯+4=35.

  • A π‘₯ = 1
  • B π‘₯ = 1 7
  • C π‘₯ = 7
  • D π‘₯ = βˆ’ 2 3 9
  • E π‘₯ = βˆ’ 1 7 9

Q14:

Solve 125π‘₯+3=5.

  • A π‘₯ = 6 2 2
  • B π‘₯ = βˆ’ 2 2
  • C π‘₯ = 2 5
  • D π‘₯ = 2 2
  • E π‘₯ = 6 2 8

Q15:

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

  • A βˆ’ 5 2 2 3
  • B 9 4
  • C 3 6 4 3
  • D 5 2 2 3

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.

  • A 2 hours and 30 minutes
  • B 2 hours and 45 minutes
  • C 3 hours and 15 minutes
  • D 3 hours and 45 minutes
  • E 3 hours and 20 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?

  • A π‘₯ βˆ’ π‘₯ βˆ’ 6 = 0 
  • B 1 2 = 0
  • C π‘₯ βˆ’ 3 = 0
  • D 0 = 0
  • E π‘₯ + 3 = 0

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

  • Athere is only one solution
  • Bthere are two solutions
  • Cthere are no solutions
  • Devery value of π‘₯ is a 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 { βˆ’ 6 , βˆ’ 2 }
  • B  βˆ’ 3 , 2 3 
  • C  βˆ’ 3 , βˆ’ 2 3 
  • D  3 , 2 3 
  • E { 6 , 2 }

Q23:

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

  • A { βˆ’ 4 , 5 }
  • B  βˆ’ 4 , βˆ’ 5 2 
  • C  βˆ’ 4 , 5 2 
  • D { 4 , βˆ’ 5 }
  • E  4 , βˆ’ 5 2 

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