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Rational Expressions Practice Tests

These Grade 11 Math practice tests develop rational-expression skills across domains, restrictions, factoring, simplification, multiplication, division, addition, subtraction, complex fractions, rational equations, asymptotes, holes, and variation models.

Skills covered

  • Finding excluded values and domains from factored denominators
  • Simplifying rational expressions by factoring and canceling common factors without losing restrictions
  • Multiplying, dividing, adding, subtracting, and simplifying complex rational expressions
  • Solving rational equations with common denominators, checking for extraneous solutions, and interpreting asymptotes and holes
  • Applying direct variation \(y=kx\), inverse variation \(y=\frac{k}{x}\), rational graphs, units, and contextual reasonableness

How to use these practice tests

Write denominator restrictions before simplifying. Factor completely, use a common denominator or reciprocal as appropriate, and substitute candidate solutions into the original expression or equation.

For more Grade 11 Math practice, return to the Grade 11 Math category.

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Grade 11 Rational Expressions Practice Tests

Practice Grade 11 Math questions about rational-expression domains, simplification, operations, rational equations, asymptotes, holes, direct variation, and inverse variation.

Choose the best answer. Factor before canceling, record denominator restrictions, use common denominators or reciprocals, and check every solution in the original equation.

10 Questions
0/10 Questions0 %
Correct 0
Incorrect 0
Unanswered 10
1. QuestionMultiple choice

What is the x-intercept of f(x)=x−4x+1f(x)=\frac{x-4}{x+1}?

Please select an answer first.

Correct answer: A

The correct answer is x=4x=4.

  1. Identify the rational-expression structure: Determine the numerator, denominator, restrictions, factors, operation, or graph feature involved.
  2. Apply the algebraic relationship: An x-intercept occurs when the numerator is zero and the denominator is nonzero, so x - 4 = 0 gives x = 4.
  3. Check restrictions and the result: The correct answer is x=4x=4.

An x-intercept occurs when the numerator is zero and the denominator is nonzero, so x - 4 = 0 gives x = 4.

2. QuestionMultiple choice

Simplify x2+5x+6x+2\frac{x^2+5x+6}{x+2} for x≠−2x\ne-2.

Please select an answer first.

Correct answer: B

The correct answer is x+3x+3.

  1. Identify the rational-expression structure: Determine the numerator, denominator, restrictions, factors, operation, or graph feature involved.
  2. Apply the algebraic relationship: Factor x^2 + 5x + 6 as (x + 2)(x + 3), then cancel x + 2 while retaining x ≠ -2.
  3. Check restrictions and the result: The correct answer is x+3x+3.

Factor x^2 + 5x + 6 as (x + 2)(x + 3), then cancel x + 2 while retaining x ≠ -2.

3. QuestionMultiple choice

Solve 1x−2=2x+1\frac{1}{x-2}=\frac{2}{x+1}.

Please select an answer first.

Correct answer: C

The correct answer is x=5x=5.

  1. Identify the rational-expression structure: Determine the numerator, denominator, restrictions, factors, operation, or graph feature involved.
  2. Apply the algebraic relationship: Cross-multiply: x + 1 = 2x - 4, so x = 5; this value does not make either original denominator zero.
  3. Check restrictions and the result: The correct answer is x=5x=5.

Cross-multiply: x + 1 = 2x - 4, so x = 5; this value does not make either original denominator zero.

4. QuestionMultiple choice

What is the vertical asymptote of f(x)=2x+1x+5f(x)=\frac{2x+1}{x+5}?

Please select an answer first.

Correct answer: D

The correct answer is x=−5x=-5.

  1. Identify the rational-expression structure: Determine the numerator, denominator, restrictions, factors, operation, or graph feature involved.
  2. Apply the algebraic relationship: The denominator is zero at x = -5, and no factor cancels with the numerator.
  3. Check restrictions and the result: The correct answer is x=−5x=-5.

The denominator is zero at x = -5, and no factor cancels with the numerator.

5. QuestionMultiple choice

What is the horizontal asymptote of f(x)=3x2−1x2+4f(x)=\frac{3x^2-1}{x^2+4}?

Please select an answer first.

Correct answer: A

The correct answer is y=3y=3.

  1. Identify the rational-expression structure: Determine the numerator, denominator, restrictions, factors, operation, or graph feature involved.
  2. Apply the algebraic relationship: The degrees are equal, so the horizontal asymptote is the ratio of leading coefficients, 3/1 = 3.
  3. Check restrictions and the result: The correct answer is y=3y=3.

The degrees are equal, so the horizontal asymptote is the ratio of leading coefficients, 3/1 = 3.

6. QuestionMultiple choice

Solve 2x+1=1\frac{2}{x+1}=1.

Please select an answer first.

Correct answer: B

The correct answer is x=1x=1.

  1. Identify the rational-expression structure: Determine the numerator, denominator, restrictions, factors, operation, or graph feature involved.
  2. Apply the algebraic relationship: Multiply by x + 1, with x ≠ -1: 2 = x + 1, so x = 1.
  3. Check restrictions and the result: The correct answer is x=1x=1.

Multiply by x + 1, with x ≠ -1: 2 = x + 1, so x = 1.

7. QuestionMultiple choice

Which value is excluded from x2−4x−2\frac{x^2-4}{x-2} even after simplification?

Please select an answer first.

Correct answer: C

The correct answer is x=2x=2.

  1. Identify the rational-expression structure: Determine the numerator, denominator, restrictions, factors, operation, or graph feature involved.
  2. Apply the algebraic relationship: The original denominator is zero at x = 2, so the simplified expression has a removable discontinuity there.
  3. Check restrictions and the result: The correct answer is x=2x=2.

The original denominator is zero at x = 2, so the simplified expression has a removable discontinuity there.

8. QuestionMultiple choice

Simplify 1/x2/x2\frac{1/x}{2/x^2} for x≠0x\ne0.

Please select an answer first.

Correct answer: D

The correct answer is x2\frac{x}{2}.

  1. Identify the rational-expression structure: Determine the numerator, denominator, restrictions, factors, operation, or graph feature involved.
  2. Apply the algebraic relationship: Multiply by the reciprocal: (1/x)(x^2/2) = x/2, with x ≠ 0 retained.
  3. Check restrictions and the result: The correct answer is x2\frac{x}{2}.

Multiply by the reciprocal: (1/x)(x^2/2) = x/2, with x ≠ 0 retained.

9. QuestionMultiple choice

Which values are excluded from x+1x2−16\frac{x+1}{x^2-16}?

Please select an answer first.

Correct answer: A

The correct answer is x=−4x=-4 and x=4x=4.

  1. Identify the rational-expression structure: Determine the numerator, denominator, restrictions, factors, operation, or graph feature involved.
  2. Apply the algebraic relationship: Factor x^2 - 16 as (x - 4)(x + 4), so x = 4 and x = -4 are excluded.
  3. Check restrictions and the result: The correct answer is x=−4x=-4 and x=4x=4.

Factor x^2 - 16 as (x - 4)(x + 4), so x = 4 and x = -4 are excluded.

10. QuestionMultiple choice

What is 5x−1−2x−1\frac{5}{x-1}-\frac{2}{x-1}?

Please select an answer first.

Correct answer: B

The correct answer is 3x−1\frac{3}{x-1}.

  1. Identify the rational-expression structure: Determine the numerator, denominator, restrictions, factors, operation, or graph feature involved.
  2. Apply the algebraic relationship: The denominators are the same, so subtract the numerators: (5 - 2)/(x - 1) = 3/(x - 1).
  3. Check restrictions and the result: The correct answer is 3x−1\frac{3}{x-1}.

The denominators are the same, so subtract the numerators: (5 - 2)/(x - 1) = 3/(x - 1).

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