Rational Expressions Worksheet with Answers – Algebra 2 (Free PDF)

Problem 1 Simplifying monomial fractions ★★★

Simplify 12x³y18xy². Assume x and y are nonzero.

Problem 2 Excluded values ★★★

State all excluded values for each expression.

1. x + 4x² − 9
2. 52x − 7
3. x − 1x² + x − 12

Problem 3 Simplifying by factoring ★★★

Simplify x² − 25x² − 3x − 10 and state the excluded values.

Problem 4 Multiplying rational expressions ★★★

Multiply and simplify 3x²x + 5 · x + 59x.

Problem 5 Excluded values on a number line ★★★

The denominator of x − 6x² − x − 20 factors as (x − 5)(x + 4). Mark the excluded values on the number line and write the domain.

Problem 6 Dividing rational expressions ★★★

Divide and simplify x² − 4x + 3 ÷ x + 2x² + 3x.

Problem 7 Like denominators ★★★

Combine and simplify.

1. 3x + 1x − 4 + 2x − 9x − 4
2. 7x + 2x + 6 − x − 10x + 6

Problem 8 Unlike denominators ★★★

Add 7x + 5 + 2x − 3.

Problem 9 Factoring a denominator first ★★★

Subtract 5x² − 9 − 2x − 3.

Problem 10 Adding with a quadratic denominator ★★★

Add xx² − 5x + 6 + 4x − 2.

Problem 11 A first rational equation ★★★

Solve 8x − 3 = 2x. Check the answer.

Problem 12 No solution ★★★

Solve xx − 5 + 2 = 5x − 5. Check every answer.

Problem 13 Solving with a graph ★★★

The graphs of y = 6x and y = x − 1 are shown. Use the graph to estimate the solutions of 6x = x − 1, then solve algebraically.

Problem 14 Two denominators and a quadratic ★★★

Solve 3x − 1 + 2x + 4 = 1. Give exact answers and check that neither is excluded.

Problem 15 Rejecting an extraneous solution ★★★

Solve 3x − 2 − 12x² − 4 = 1.

Problem 16 Painting a fence together ★★★

A robotics club paints a fence for its field. Ana alone needs 6 hours and Ben alone needs 9 hours. How long do they need working together? The figure shows their hourly shares of the job.

Problem 17 Two hoses filling a pond ★★★

Two hoses together fill a pond in 6 hours. The slower hose alone takes 5 hours longer than the faster hose alone. How long does each hose take alone?

Problem 18 Paddling against a current ★★★

A kayaker paddles 12 miles upstream and then 12 miles back downstream in a river with a 3 mph current. The round trip takes 3 hours. Find the kayaker’s speed in still water.

Problem 19 Resistors in parallel ★★★

Two resistors in parallel have resistances x ohms and x + 6 ohms. The combined resistance is 4 ohms, where 1R = 1R₁ + 1R₂. Find both resistances.

Problem 20 A multi-step quotient ★★★

Simplify x² − 9x² + 2x ÷ x + 3x² − 4 and list every excluded value.