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Solving Rational Equations

Equations with rational expressions

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Solving Rational Equations

Strategy

  1. Find the LCD of all denominators
  2. Multiply both sides by the LCD
  3. Solve the resulting equation
  4. Check for extraneous solutions

Extraneous Solutions

Solutions that make any denominator zero are extraneous and must be rejected.

Always check your answers!

Common Types

Proportion: ab=cd\frac{a}{b} = \frac{c}{d} → Cross multiply: ad=bcad = bc

Work Problems: 1t1+1t2=1ttogether\frac{1}{t_1} + \frac{1}{t_2} = \frac{1}{t_{together}}

Rate Problems: Time=DistanceRate\text{Time} = \frac{\text{Distance}}{\text{Rate}}

Example

Solve: 3x+2x−1=5\frac{3}{x} + \frac{2}{x-1} = 5

LCD: x(x−1)x(x - 1)

Multiply both sides: 3(x−1)+2x=5x(x−1)3(x - 1) + 2x = 5x(x - 1) 3x−3+2x=5x2−5x3x - 3 + 2x = 5x^2 - 5x 5x−3=5x2−5x5x - 3 = 5x^2 - 5x 0=5x2−10x+30 = 5x^2 - 10x + 3

Use quadratic formula to solve.

📚 Practice Problems

1Problem 1easy

❓ Question:

Solve: x3=52\frac{x}{3} = \frac{5}{2}

💡 Show Solution

Cross multiply: 2x=3⋅52x = 3 \cdot 5 2x=152x = 15 x=152x = \frac{15}{2}

Check: 15/23=156=52\frac{15/2}{3} = \frac{15}{6} = \frac{5}{2} ✓

Answer: x=152x = \frac{15}{2} or 7.57.5

2Problem 2medium

❓ Question:

Solve: 1x+1x+2=12\frac{1}{x} + \frac{1}{x+2} = \frac{1}{2}

💡 Show Solution

LCD: 2x(x+2)2x(x + 2)

Multiply both sides by LCD: 2x(x+2)⋅1x+2x(x+2)⋅1x+2=2x(x+2)⋅122x(x + 2) \cdot \frac{1}{x} + 2x(x + 2) \cdot \frac{1}{x+2} = 2x(x + 2) \cdot \frac{1}{2}

2(x+2)+2x=x(x+2)2(x + 2) + 2x = x(x + 2) 2x+4+2x=x2+2x2x + 4 + 2x = x^2 + 2x 4x+4=x2+2x4x + 4 = x^2 + 2x 0=x2−2x−40 = x^2 - 2x - 4

Use quadratic formula: a=1,b=−2,c=−4a = 1, b = -2, c = -4 x=2±4+162=2±202=2±252=1±5x = \frac{2 \pm \sqrt{4 + 16}}{2} = \frac{2 \pm \sqrt{20}}{2} = \frac{2 \pm 2\sqrt{5}}{2} = 1 \pm \sqrt{5}

Answer: x=1+5x = 1 + \sqrt{5} or x=1−5x = 1 - \sqrt{5}

3Problem 3hard

❓ Question:

Solve: 6x−2=xx−2+1\frac{6}{x-2} = \frac{x}{x-2} + 1

💡 Show Solution

LCD: x−2x - 2

Multiply both sides: 6=x+(x−2)6 = x + (x - 2) 6=x+x−26 = x + x - 2 6=2x−26 = 2x - 2 8=2x8 = 2x x=4x = 4

Check: Does x=4x = 4 make any denominator zero? x−2=4−2=2≠0x - 2 = 4 - 2 = 2 \neq 0 ✓

Verify: 62=42+1\frac{6}{2} = \frac{4}{2} + 1 → 3=2+13 = 2 + 1 ✓

Answer: x=4x = 4

Explain using:

⚠️ Common Mistakes: Solving Rational Equations

Avoid these 3 frequent errors

🌍 Real-World Applications: Solving Rational Equations

See how this math is used in the real world

📝 Worked Example: Solving a Quadratic by Factoring

Problem:

Solve x2−5x+6=0x^2 - 5x + 6 = 0.

2Factor the quadratic
3Set each factor equal to zero
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❓ Frequently Asked Questions

What is Solving Rational Equations?▾
Equations with rational expressions
How can I study Solving Rational Equations effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 3 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Solving Rational Equations study guide free?▾
Yes — all study notes, flashcards, and practice problems for Solving Rational Equations on Study Mondo are free to access. No account is needed.
What course covers Solving Rational Equations?▾
Solving Rational Equations is part of the Algebra 2 course on Study Mondo, specifically in the Rational Expressions section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Solving Rational Equations?▾
Yes, this page includes 3 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.