Worksheet: Adding and Subtracting Rational Expressions

In this worksheet, we will practice adding and subtracting rational expressions, identifying the domain of the resulting functions, and simplifying them.

Q1:

Simplify the function 𝑛 ( 𝑥 ) = 8 𝑥 6 + 𝑥 6 𝑥 6 𝑥 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 𝑥 1 4 𝑥 ( 𝑥 6 ) , domain = { 0 , 6 }
  • B 𝑛 ( 𝑥 ) = 7 𝑥 + 6 𝑥 ( 𝑥 6 ) , domain = { 0 , 6 }
  • C 𝑛 ( 𝑥 ) = 𝑥 1 4 𝑥 ( 𝑥 6 ) , domain = { 0 , 6 }
  • D 𝑛 ( 𝑥 ) = 7 𝑥 + 6 𝑥 ( 𝑥 6 ) , domain = { 0 , 6 }
  • E 𝑛 ( 𝑥 ) = 𝑥 1 4 𝑥 5 𝑥 6 2 , domain = { 0 , 6 }

Q2:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 7 𝑥 3 𝑥 2 8 𝑥 7 7 𝑥 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 4 , domain = { 4 , 7 }
  • B 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 + 4 , domain = { 4 }
  • C 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 4 , domain = { 4 }
  • D 𝑛 ( 𝑥 ) = 𝑥 + 5 𝑥 + 4 , domain = { 4 , 7 }
  • E 𝑛 ( 𝑥 ) = 2 𝑥 + 4 , domain = { 4 , 7 }

Q3:

Answer the following questions for the rational expressions 𝑥 + 3 3 and 𝑥 8 2 𝑥 .

Find the sum of 𝑥 + 3 3 and 𝑥 8 2 𝑥 .

  • A 𝑥 + 9 𝑥 2 4 6 𝑥
  • B 2 𝑥 + 3 𝑥 + 5 𝑥
  • C 2 𝑥 + 3 𝑥 5 𝑥
  • D 2 𝑥 + 9 𝑥 2 4 6 𝑥
  • E 𝑥 + 9 𝑥 2 4 6 𝑥

Is the sum of 𝑥 + 3 3 and 𝑥 8 2 𝑥 a rational expression?

  • A yes
  • B no

Would this be true for any two rational expressions summed together?

  • A no
  • B yes

Q4:

Simplify the function 𝑛 ( 𝑥 ) = 7 𝑥 𝑥 1 + 3 𝑥 1 𝑥 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 7 𝑥 + 3 𝑥 ( 𝑥 1 ) ( 1 𝑥 ) 2 , domain = { 1 }
  • B 𝑛 ( 𝑥 ) = 𝑥 ( 7 𝑥 3 ) 𝑥 1 , domain = { 1 , 1 }
  • C 𝑛 ( 𝑥 ) = 7 𝑥 + 3 𝑥 ( 𝑥 1 ) ( 1 𝑥 ) 2 , domain = { 1 , 1 }
  • D 𝑛 ( 𝑥 ) = 𝑥 ( 7 𝑥 3 ) 𝑥 1 , domain = { 1 }
  • E 𝑛 ( 𝑥 ) = 𝑥 ( 7 𝑥 + 3 ) 𝑥 1 , domain = { 1 }

Q5:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 + 7 𝑥 + 6 + 𝑥 3 𝑥 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 4 𝑥 + 1 3 3 𝑥 , domain = { 0 }
  • B 𝑛 ( 𝑥 ) = 4 𝑥 + 2 7 3 𝑥 , domain = { 0 , 3 }
  • C 𝑛 ( 𝑥 ) = 4 𝑥 + 1 3 3 𝑥 , domain = { 0 , 3 }
  • D 𝑛 ( 𝑥 ) = 4 𝑥 + 2 7 3 𝑥 , domain = { 0 }
  • E 𝑛 ( 𝑥 ) = 2 𝑥 + 1 3 4 𝑥 , domain = { 0 }

Q6:

Simplify the function 𝑛 ( 𝑥 ) = 3 𝑥 𝑥 + 4 7 𝑥 𝑥 4 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 4 𝑥 ( 𝑥 + 4 ) ( 𝑥 4 ) , domain = { 4 , 4 }
  • B 𝑛 ( 𝑥 ) = 4 𝑥 ( 𝑥 + 1 0 ) ( 𝑥 + 4 ) ( 𝑥 4 ) , domain = { 4 , 4 }
  • C 𝑛 ( 𝑥 ) = 4 𝑥 ( 𝑥 + 4 ) ( 𝑥 4 ) , domain = { 4 , 4 }
  • D 𝑛 ( 𝑥 ) = 4 𝑥 ( 𝑥 + 1 0 ) ( 𝑥 + 4 ) ( 𝑥 4 ) , domain = { 4 , 4 }
  • E 𝑛 ( 𝑥 ) = 4 𝑥 ( 𝑥 1 0 ) ( 𝑥 + 4 ) ( 𝑥 4 ) , domain = { 4 , 4 }

Q7:

Given that the domain of the function 𝑛 ( 𝑥 ) = 𝑏 𝑥 + 6 𝑥 + 𝑎 is { 4 , 0 } , and 𝑛 ( 1 ) = 2 , find the values of 𝑎 and 𝑏 .

  • A 𝑎 = 4 , 𝑏 = 2
  • B 𝑎 = 4 , 𝑏 = 0
  • C 𝑎 = 4 , 𝑏 = 2
  • D 𝑎 = 4 , 𝑏 = 0
  • E 𝑎 = 4 , 𝑏 = 4

Q8:

Simplify the function 𝑛 ( 𝑥 ) = 8 𝑥 + 7 𝑥 1 4 𝑥 + 4 5 + 3 𝑥 2 4 𝑥 1 7 𝑥 + 7 2 2 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 1 1 𝑥 1 7 2 𝑥 3 1 𝑥 + 1 1 7 2 , domain = { 5 , 8 , 9 }
  • B 𝑛 ( 𝑥 ) = 1 1 𝑥 8 ( 𝑥 9 ) ( 𝑥 5 ) , domain = { 5 , 9 }
  • C 𝑛 ( 𝑥 ) = 2 ( 4 𝑥 + 5 ) ( 𝑥 9 ) ( 𝑥 5 ) , domain = { 5 , 8 , 9 }
  • D 𝑛 ( 𝑥 ) = 1 1 𝑥 8 ( 𝑥 9 ) ( 𝑥 5 ) , domain = { 5 , 8 , 9 }
  • E 𝑛 ( 𝑥 ) = 2 ( 4 𝑥 + 5 ) ( 𝑥 9 ) ( 𝑥 5 ) , domain = { 5 , 9 }

Q9:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 + 𝑥 + 1 𝑥 1 8 1 𝑥 𝑥 1 0 𝑥 + 9 2 3 2 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 𝑥 8 𝑥 1 , domain = { 1 , 9 }
  • B 𝑛 ( 𝑥 ) = 𝑥 + 1 0 𝑥 1 , domain = { 1 }
  • C 𝑛 ( 𝑥 ) = 𝑥 8 𝑥 1 , domain = { 1 }
  • D 𝑛 ( 𝑥 ) = 𝑥 + 1 0 𝑥 1 , domain = { 1 , 9 }
  • E 𝑛 ( 𝑥 ) = 𝑥 8 𝑥 1 , domain = { 1 , 9 }

Q10:

Answer the following questions for the rational expressions 5 𝑥 2 3 𝑥 and 3 𝑥 2 𝑥 2 𝑥 + 8 .

Subtract 5 𝑥 2 3 𝑥 from 3 𝑥 2 𝑥 2 𝑥 + 8 .

  • A 𝑥 3 𝑥 + 3 6 𝑥 𝑥 ( 𝑥 + 4 )
  • B 𝑥 + 2 𝑥 3 6 𝑥 + 1 6 6 𝑥 ( 𝑥 + 4 )
  • C 9 𝑥 2 𝑥 3 6 𝑥 + 1 6 6 𝑥 ( 𝑥 + 4 )
  • D 9 𝑥 1 6 𝑥 3 6 𝑥 + 1 6 6 𝑥 ( 𝑥 + 4 )
  • E 𝑥 1 6 𝑥 + 3 6 𝑥 𝑥 ( 𝑥 + 4 )

Is the difference between 3 𝑥 2 𝑥 2 𝑥 + 8 and 5 𝑥 2 3 𝑥 a rational expression?

  • A yes
  • B no

Is the result of this subtraction a rational expression?

  • A no
  • B yes

Q11:

Simplify the function 𝑛 ( 𝑥 ) = 9 𝑥 + 6 + 9 𝑥 6 , and determine its domain in .

  • A 𝑛 ( 𝑥 ) = 1 8 ( 𝑥 6 ) ( 𝑥 + 6 ) , domain = { 6 , 6 }
  • B 𝑛 ( 𝑥 ) = 1 8 𝑥 ( 𝑥 6 ) ( 𝑥 + 6 ) , domain = { 6 }
  • C 𝑛 ( 𝑥 ) = 9 𝑥 , domain = { 6 , 6 }
  • D 𝑛 ( 𝑥 ) = 1 8 𝑥 ( 𝑥 6 ) ( 𝑥 + 6 ) , domain = { 6 , 6 }

Q12:

Simplify the function 𝑛 ( 𝑥 ) = 5 𝑥 𝑥 4 𝑥 + 4 𝑥 1 6 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 5 𝑥 + 1 𝑥 4 , domain = { 4 , 4 }
  • B 𝑛 ( 𝑥 ) = 5 𝑥 1 𝑥 4 , domain = { 4 }
  • C 𝑛 ( 𝑥 ) = 5 𝑥 + 1 𝑥 4 , domain = { 4 }
  • D 𝑛 ( 𝑥 ) = 5 𝑥 1 𝑥 4 , domain = { 4 , 4 }
  • E 𝑛 ( 𝑥 ) = 4 ( 𝑥 1 ) ( 𝑥 4 ) ( 𝑥 1 6 ) 2 , domain = { 4 , 4 }

Q13:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 + 3 𝑥 4 9 3 𝑥 + 2 1 𝑥 3 𝑥 6 𝑥 1 0 5 𝑥 2 3 2 4 3 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 2 ( 1 1 𝑥 + 3 2 ) ( 𝑥 7 ) ( 𝑥 + 7 ) ( 𝑥 + 5 ) , domain = { 7 , 5 , 0 , 7 }
  • B 𝑛 ( 𝑥 ) = 2 ( 3 𝑥 + 1 7 ) ( 𝑥 7 ) ( 𝑥 + 7 ) ( 𝑥 + 5 ) , domain = { 7 , 5 , 7 }
  • C 𝑛 ( 𝑥 ) = 2 ( 1 1 𝑥 + 3 2 ) ( 𝑥 7 ) ( 𝑥 + 7 ) ( 𝑥 + 5 ) , domain = { 7 , 5 , 7 }
  • D 𝑛 ( 𝑥 ) = 2 ( 3 𝑥 + 1 7 ) ( 𝑥 7 ) ( 𝑥 + 7 ) ( 𝑥 + 5 ) , domain = { 7 , 5 , 0 , 7 }
  • E 𝑛 ( 𝑥 ) = 3 𝑥 2 1 𝑥 + 𝑥 + 3 ( 𝑥 7 ) ( 𝑥 + 7 ) ( 𝑥 + 5 ) 3 2 , domain = { 7 , 5 , 0 , 7 }

Q14:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 + 1 3 𝑥 𝑥 3 𝑥 8 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 1 8 , domain = 0 , 8 3
  • B 𝑛 ( 𝑥 ) = 5 𝑥 + 8 3 𝑥 ( 3 𝑥 8 ) , domain = 0 , 8 3
  • C 𝑛 ( 𝑥 ) = 1 8 , domain =
  • D 𝑛 ( 𝑥 ) = 5 𝑥 + 8 3 𝑥 ( 3 𝑥 8 ) , domain = 0 , 8 3

Q15:

Simplify the function 𝑛 ( 𝑥 ) = ( 𝑥 8 ) 𝑥 𝑥 + 8 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 6 4 𝑥 + 8 , domain = { 8 }
  • B 𝑛 ( 𝑥 ) = 6 4 𝑥 + 8 , domain = { 8 }
  • C 𝑛 ( 𝑥 ) = 𝑥 + 𝑥 8 𝑥 + 8 2 , domain = { 8 }
  • D 𝑛 ( 𝑥 ) = 6 4 𝑥 + 8 , domain = { 8 }
  • E 𝑛 ( 𝑥 ) = 6 4 𝑥 + 8 , domain = { 8 }

Q16:

Simplify the function 𝑛 ( 𝑥 ) = 6 4 𝑥 1 + 9 9 𝑥 1 8 𝑥 2 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 1 5 1 4 𝑥 + 9 𝑥 1 2 , domain = 1 2 , 0 , 1 2
  • B 𝑛 ( 𝑥 ) = 4 𝑥 1 𝑥 ( 2 𝑥 + 1 ) ( 2 𝑥 1 ) , domain = 1 2 , 1 2
  • C 𝑛 ( 𝑥 ) = 5 𝑥 ( 2 𝑥 + 1 ) ( 2 𝑥 1 ) , domain = 1 2 , 0 , 1 2
  • D 𝑛 ( 𝑥 ) = 4 𝑥 1 𝑥 ( 2 𝑥 + 1 ) ( 2 𝑥 1 ) , domain = 1 2 , 0 , 1 2

Q17:

Simplify the function 𝑛 ( 𝑥 ) = 5 𝑥 + 8 + 7 𝑥 + 4 6 4 𝑥 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 4 ( 3 𝑥 + 1 1 ) ( 𝑥 8 ) ( 𝑥 + 8 ) , domain = { 8 , 8 }
  • B 𝑛 ( 𝑥 ) = 1 2 ( 𝑥 3 ) ( 𝑥 8 ) ( 𝑥 + 8 ) , domain = { 8 }
  • C 𝑛 ( 𝑥 ) = 4 ( 3 𝑥 + 1 1 ) ( 𝑥 8 ) ( 𝑥 + 8 ) , domain = { 8 }
  • D 𝑛 ( 𝑥 ) = 1 2 ( 𝑥 3 ) ( 𝑥 8 ) ( 𝑥 + 8 ) , domain = { 8 , 8 }
  • E 𝑛 ( 𝑥 ) = 1 2 ( 𝑥 + 3 ) ( 𝑥 8 ) ( 𝑥 + 8 ) , domain = { 8 , 8 }

Q18:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 6 8 𝑥 5 7 𝑥 + 5 4 + 𝑥 5 8 1 𝑥 8 1 8 𝑥 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 1 4 ( 8 𝑥 9 ) ( 𝑥 9 ) , domain = 9 8 , 6 , 9
  • B 𝑛 ( 𝑥 ) = 4 ( 8 𝑥 9 ) ( 𝑥 9 ) , domain = 9 8 , 9
  • C 𝑛 ( 𝑥 ) = 1 4 ( 8 𝑥 9 ) ( 𝑥 9 ) , domain = 9 8 , 9
  • D 𝑛 ( 𝑥 ) = 4 ( 8 𝑥 9 ) ( 𝑥 9 ) , domain = 9 8 , 6 , 9
  • E 𝑛 ( 𝑥 ) = 1 4 ( 8 𝑥 + 9 ) ( 𝑥 + 9 ) , domain = 9 8 , 6 , 9

Q19:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 4 𝑥 3 2 𝑥 1 6 𝑥 + 6 4 + 𝑥 8 𝑥 + 1 2 𝑥 1 0 𝑥 + 1 6 2 2 2 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 2 ( 𝑥 + 1 ) 𝑥 8 , domain = { 2 , 8 }
  • B 𝑛 ( 𝑥 ) = 2 ( 𝑥 1 ) 𝑥 8 , domain = { 8 }
  • C 𝑛 ( 𝑥 ) = 2 ( 𝑥 + 1 ) 𝑥 8 , domain = { 8 }
  • D 𝑛 ( 𝑥 ) = 2 ( 𝑥 1 ) 𝑥 8 , domain = { 2 , 8 }
  • E 𝑛 ( 𝑥 ) = 2 ( 𝑥 1 ) 𝑥 + 8 , domain = { 2 , 8 }

Q20:

Simplify the function 𝑛 ( 𝑥 ) = 𝑥 7 6 𝑥 3 5 𝑥 4 9 + 𝑥 + 2 1 4 + 1 9 𝑥 + 6 𝑥 2 2 , and determine its domain.

  • A 𝑛 ( 𝑥 ) = 2 6 𝑥 7 , domain = 7 , 2 , 7 6
  • B 𝑛 ( 𝑥 ) = 2 6 𝑥 + 7 , domain = 7 6
  • C 𝑛 ( 𝑥 ) = 2 6 𝑥 7 , domain = 7 6
  • D 𝑛 ( 𝑥 ) = 2 6 𝑥 + 7 , domain = 7 , 2 , 7 6
  • E 𝑛 ( 𝑥 ) = 2 𝑥 5 1 2 𝑥 1 6 𝑥 3 5 2 , domain = 7 , 2 , 7 6

Q21:

Given that the domain of the function 𝑛 ( 𝑥 ) = 2 𝑥 ( 𝑥 𝑎 ) ( 𝑥 + 6 ) + 5 𝑥 + 5 ( 𝑥 𝑎 ) ( 𝑥 + 3 ) is { 6 , 3 , 2 } , what is the value of 𝑎 ?

  • A { 2 }
  • B { 2 }
  • C { 2 }
  • D { 2 }
  • E { 6 , 3 , 2 }

Q22:

Simplify the function 𝑛 ( 𝑥 ) = 4 𝑥 8 𝑥 + 2 𝑥 8 + 2 𝑥 + 4 𝑥 + 4 × 𝑥 + 2 7 𝑥 3 𝑥 + 9 , and find its domain.

  • A 𝑛 ( 𝑥 ) = 2 ( 𝑥 3 ) , domain = { 4 , 2 }
  • B 𝑛 ( 𝑥 ) = 2 ( 𝑥 + 3 ) , domain = { 4 , 3 , 2 }
  • C 𝑛 ( 𝑥 ) = 2 ( 𝑥 3 ) , domain = { 4 , 3 , 2 }
  • D 𝑛 ( 𝑥 ) = 2 ( 𝑥 + 3 ) , domain = { 4 , 2 }
  • E 𝑛 ( 𝑥 ) = 2 𝑥 + 3 , domain = { 4 , 2 }

Q23:

Simplify 5 𝑥 2 3 𝑥 7 𝑥 2 9 𝑥 2 4 .

  • A 2 3 𝑥 9 𝑥 3
  • B 2 9 𝑥 3
  • C 5 𝑥 6 𝑥 + 7 𝑥 9 𝑥 3 2 4
  • D 1 5 𝑥 6 𝑥 7 𝑥 + 2 9 𝑥 3 2 4
  • E 1 5 𝑥 6 𝑥 7 𝑥 + 2 9 𝑥 3 2 4

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