How to Solve Rational Equations? (+FREE Worksheet!)
To solve a rational equation, multiply through by the least common denominator to clear the fractions, solve the resulting equation, then check every answer against the original. Any value that makes a denominator zero is extraneous and must be discarded.
Tutor-style math help
Solve Rational Equations: what to notice and how to work it
Rational skill
Rational expressions are algebraic fractions. Restrictions matter from the beginning because a denominator can never be zero.
What to notice first
Factor before simplifying. You may cancel common factors, but you may not cancel pieces of sums.
Common student mistake
Do not cancel terms across plus or minus signs. In \((x+2)/x\), the x in the denominator is not a common factor of the entire numerator.
Key formulas and cues
\(\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}\)
\(\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\)
\(\text{denominator}\ne0\)
\(\text{vertical asymptote: denominator}=0\text{ after simplification checks}\)
A reliable path
- State restrictionsFind values that make original denominators zero.
- Factor and simplifyCancel only factors shared by the whole numerator and denominator.
- Check the resultKeep original restrictions and watch for asymptotes or holes when graphing.
Worked examples
Simplify safely
Example: \(\frac{6x}{9x}\), \(x\ne0\)
- Cancel the common factor x.
- Reduce 6/9.
- Keep the restriction x not equal to 0.
Answer: \(\frac{2}{3},\ x\ne0\)
Find a restriction
Example: \(\frac{x+1}{x-4}\)
- Look at the denominator.
- Set x – 4 = 0.
- Exclude that value.
Answer: \(x\ne4\)
Try one before moving on
Try: Simplify \(\frac{x^2+3x}{x}\), \(x\ne0\).
Answer: \(x+3,\ x\ne0\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Solve Rational Equations: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. You may cancel:
2. For \(\frac{5}{x+2}\), x cannot be:
3. A vertical asymptote often comes from:
Rational Equations, Example 2:
Rational Equations, Example 3:
Rational Equations, Example 4:
Exercises for Rational Equations
Solve Rational Equations.
- \(\color{blue}{\frac{10}{x+4}=\frac{15}{4x+4}}\)
- \(\color{blue}{\frac{x+4}{x+1}=\frac{x-6}{x-1}}\)
- \(\color{blue}{\frac{2x}{x+3}=\frac{x-6}{x+4}}\)
- \(\color{blue}{\frac{1}{x+5}-1=\frac{1}{1+x}}\)
- \(\color{blue}{\frac{1}{5x^2}-\frac{1}{x}=\frac{2}{x}}\)
- \(\color{blue}{\frac{2x}{2x-2}-\frac{2}{x}=\frac{1}{x-1}}\)
- \(\color{blue}{x=\frac{4}{5}}\)
- \(\color{blue}{x=-\frac{1}{4}}\)
- \(\color{blue}{x=-9}\) or \(\color{blue}{x=-2}\)
- \(\color{blue}{x=-3}\)
- \(\color{blue}{x=\frac{1}{15}}\)
- \(\color{blue}{x=2}\)
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