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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

Solve 2x=13\frac{2}{x}=\frac{1}{3}.

Please select an answer first.

Correct answer: A

The correct answer is x=6x=6.

  1. Identify the rational-expression structure: Determine the denominators, restrictions, common factors, operation, or equation involved.
  2. Apply the algebraic relationship: Cross-multiply: 2(3) = x, so x = 6.
  3. Check restrictions and the result: The correct answer is x=6x=6.

Cross-multiply: 2(3) = x, so x = 6.

2. QuestionMultiple choice

Solve x+1x−2=3\frac{x+1}{x-2}=3.

Please select an answer first.

Correct answer: B

The correct answer is x=72x=\frac{7}{2}.

  1. Identify the rational-expression structure: Determine the denominators, restrictions, common factors, operation, or equation involved.
  2. Apply the algebraic relationship: Multiply by x - 2: x + 1 = 3x - 6, so 7 = 2x and x = 7/2.
  3. Check restrictions and the result: The correct answer is x=72x=\frac{7}{2}.

Multiply by x - 2: x + 1 = 3x - 6, so 7 = 2x and x = 7/2.

3. QuestionMultiple choice

Which value must be excluded when solving 1x−4=2x\frac{1}{x-4}=\frac{2}{x}?

Please select an answer first.

Correct answer: C

The correct answer is x=0x=0 and x=4x=4.

  1. Identify the rational-expression structure: Determine the denominators, restrictions, common factors, operation, or equation involved.
  2. Apply the algebraic relationship: Both original denominators must be nonzero, so x cannot be 0 or 4.
  3. Check restrictions and the result: The correct answer is x=0x=0 and x=4x=4.

Both original denominators must be nonzero, so x cannot be 0 or 4.

4. QuestionMultiple choice

What is an extraneous solution?

Please select an answer first.

Correct answer: D

The correct answer is A value produced by algebraic steps that does not satisfy the original equation.

  1. Identify the rational-expression structure: Determine the denominators, restrictions, common factors, operation, or equation involved.
  2. Apply the algebraic relationship: Multiplying by expressions containing variables can introduce values excluded from the original equation.
  3. Check restrictions and the result: The correct answer is A value produced by algebraic steps that does not satisfy the original equation.

Multiplying by expressions containing variables can introduce values excluded from the original equation.

5. QuestionMultiple choice

Solve xx+1=23\frac{x}{x+1}=\frac{2}{3}.

Please select an answer first.

Correct answer: A

The correct answer is x=2x=2.

  1. Identify the rational-expression structure: Determine the denominators, restrictions, common factors, operation, or equation involved.
  2. Apply the algebraic relationship: Cross-multiply: 3x = 2x + 2, so x = 2, which is allowed because x is not -1.
  3. Check restrictions and the result: The correct answer is x=2x=2.

Cross-multiply: 3x = 2x + 2, so x = 2, which is allowed because x is not -1.

6. QuestionMultiple choice

What is the simplified form of x2−9x+3\frac{x^2-9}{x+3} for x≠−3x\ne-3?

Please select an answer first.

Correct answer: B

The correct answer is x−3x-3.

  1. Identify the rational-expression structure: Recognize the numerator as a difference of squares and note that x = -3 is excluded.
  2. Apply the algebraic relationship: Factor x^2 - 9 as (x - 3)(x + 3), then cancel x + 3 while retaining the restriction x ≠ -3.
  3. Check restrictions and the result: The simplified expression is x - 3, but the original restriction x ≠ -3 remains.

Factor x^2 - 9 as (x - 3)(x + 3), then cancel x + 3 while retaining the restriction x ≠ -3.

7. QuestionMultiple choice

Solve 3x+2=1x−1\frac{3}{x+2}=\frac{1}{x-1}.

Please select an answer first.

Correct answer: C

The correct answer is x=52x=\frac{5}{2}.

  1. Identify the rational-expression structure: Determine the denominators, restrictions, common factors, operation, or equation involved.
  2. Apply the algebraic relationship: Cross-multiplication gives 3(x - 1) = x + 2, so 3x - 3 = x + 2, 2x = 5, and x = 5/2; this value is not an excluded denominator.
  3. Check restrictions and the result: The correct answer is x=52x=\frac{5}{2}.

Cross-multiplication gives 3(x - 1) = x + 2, so 3x - 3 = x + 2, 2x = 5, and x = 5/2; this value is not an excluded denominator.

8. QuestionMultiple choice

What is the vertical asymptote of f(x)=x+1x−3f(x)=\frac{x+1}{x-3}?

Please select an answer first.

Correct answer: D

The correct answer is x=3x=3.

  1. Identify the rational-expression structure: Determine the denominators, restrictions, common factors, operation, or equation involved.
  2. Apply the algebraic relationship: The denominator is zero at x = 3 and the factor does not cancel.
  3. Check restrictions and the result: The correct answer is x=3x=3.

The denominator is zero at x = 3 and the factor does not cancel.

9. QuestionMultiple choice

What is the horizontal asymptote of f(x)=4x2+12x2−3f(x)=\frac{4x^2+1}{2x^2-3}?

Please select an answer first.

Correct answer: A

The correct answer is y=2y=2.

  1. Identify the rational-expression structure: Determine the denominators, restrictions, common factors, operation, or equation involved.
  2. Apply the algebraic relationship: Equal degrees give the ratio of leading coefficients, 4/2 = 2.
  3. Check restrictions and the result: The correct answer is y=2y=2.

Equal degrees give the ratio of leading coefficients, 4/2 = 2.

10. QuestionMultiple choice

What does a removable discontinuity usually appear as on a rational graph?

Please select an answer first.

Correct answer: B

The correct answer is A hole.

  1. Identify the rational-expression structure: Determine the denominators, restrictions, common factors, operation, or equation involved.
  2. Apply the algebraic relationship: A common factor can create a missing point after cancellation, producing a hole rather than an infinite discontinuity.
  3. Check restrictions and the result: The correct answer is A hole.

A common factor can create a missing point after cancellation, producing a hole rather than an infinite discontinuity.

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