Worksheet: Rational Equations by Cross Multiplication

In this worksheet, we will practice solving rational equations using cross multiplication.

Q1:

Solve 1 2 5 π‘₯ + 3 = 5 .

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

Q2:

Solve 2 π‘₯ + 1 3 π‘₯ + 4 = 3 5 .

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

Q3:

If the average of two numbers is 65, where one of them is 60, find the other number.

  • A190
  • B5
  • C92
  • D70

Q4:

Find π‘₯ given π‘₯ βˆ’ 2 0 π‘₯ + 1 0 = 0 .

Q5:

Given that 𝑛 ( π‘₯ ) = 7 + 𝑏 π‘₯ βˆ’ 7  , 𝑛 ( π‘₯ ) = 2 π‘₯ βˆ’ 7  , and 𝑛 ( π‘₯ ) = 𝑛 ( π‘₯ )   , what is the value of 𝑏 ?

Q6:

Consider the function 𝑓 ( π‘₯ ) = 6 π‘₯ + 8 π‘₯ + 1 2 .

For what values of 𝑐 does 𝑓 ( π‘₯ ) = 𝑐 have a solution?

  • A βˆ’ 1 and 9
  • Ball real numbers except βˆ’ 1 and 9
  • C 𝑐 ≀ 9
  • D βˆ’ 1 ≀ 𝑐 ≀ 9
  • Eall real numbers

What is the range of the function 𝑓 ( π‘₯ ) ?

  • A βˆ’ 1 ≀ 𝑐 ≀ 9
  • B βˆ’ 9 ≀ 𝑐 ≀ 1
  • Call real numbers
  • Dall real numbers except βˆ’ 1 and 9
  • E 1 ≀ 𝑐 ≀ 9

Q7:

If 2 π‘₯ + 7 5 π‘₯ βˆ’ 9 = 1 2 , find the value of π‘₯ .

Q8:

Given that π‘₯ + 𝑦 4 5 = 𝑦 + 𝑧 3 3 , which of the following is equal to π‘₯ βˆ’ 𝑧 2 ?

  • A π‘₯ + 2 𝑦 + 𝑧 2
  • B π‘₯ + 2 𝑦 + 𝑧 7 8
  • C π‘₯ + 2 𝑦 + 𝑧 1 2
  • D π‘₯ + 2 𝑦 + 𝑧 1 3

Q9:

Container 𝐿 has one litre of acid at a 2 5 % concentration, and container 𝐻 has the same acid at a 7 5 % concentration. How many litres from 𝐻 must be added to 𝐿 to obtain a 6 7 % concentration?

  • A 7.35 litres
  • B 2.17 litres
  • C 4.28 litres
  • D 5.25 litres
  • E 6.97 litres

Q10:

Given that the ratio ο€Ή π‘₯  βˆ’ 8  ∢ ο€Ή 5 π‘₯  + 6  is equivalent to 7 ∢ 3 7 , then what are the possible values of π‘₯ ?

  • A 1 1 , βˆ’ 1 1
  • B 3 , βˆ’ 3
  • C 1 6 , βˆ’ 1 6
  • D 1 3 , βˆ’ 1 3

Q11:

If 9 π‘₯ βˆ’ 3 𝑦 7 π‘₯ + 𝑦 = 1 3 1 9 , find π‘₯ ∢ 𝑦 in its simplest form.

  • A 2 9 ∢ 8
  • B 8 ∢ 7
  • C 8 ∢ 2 9
  • D 7 ∢ 8
  • E 1 1 ∢ 2 0

Q12:

If the function 𝑛 ( π‘₯ ) = π‘₯ + π‘₯ π‘₯ βˆ’ 1   , what is the value of π‘₯ for which the multiplicative inverse of 𝑛 is 16?

  • A 1 1 5
  • B 1 5 1 6
  • C βˆ’ 1 5 1 6
  • D βˆ’ 1 1 5
  • E 1 1 7

Q13:

Given that 𝑓 ∢ ℝ βˆ’ { βˆ’ 1 } β†’ ℝ , where 𝑓 ( π‘₯ ) = π‘₯ + π‘Ž π‘₯ βˆ’ 𝑏 and 𝑓 ( βˆ’ 5 ) = βˆ’ 1 4 , determine the values of π‘Ž and 𝑏 .

  • A π‘Ž = βˆ’ 9 2 , 𝑏 = 1
  • B π‘Ž = 1 , 𝑏 = βˆ’ 2 1
  • C π‘Ž = βˆ’ 5 , 𝑏 = βˆ’ 1
  • D π‘Ž = 6 , 𝑏 = βˆ’ 1
  • E π‘Ž = βˆ’ 1 , 𝑏 = βˆ’ 2 9

Q14:

Find the solution set of 1 ( π‘₯ βˆ’ 1 2 ) = 0 . 0 0 0 1 4 in ℝ .

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

Q15:

Find the value of π‘₯ given π‘₯ + 4 4 π‘₯ is the multiplicative inverse of 9 1 3 .

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

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