Rational Equations
Multiply by the LCD to clear denominators; check extraneous solutions.
Key idea
Multiply by the LCD to clear denominators; check extraneous solutions.
In depth
Multiply every term by the least common denominator to clear fractions, solve, and then check each answer. Some solutions make a denominator zero and must be rejected (extraneous).
Worked example
1/x + 1/2 = 1: multiply by 2x → 2 + x = 2x → x = 2. Check: 1/2 + 1/2 = 1. ✓
Real-world uses
- Solving combined work problems, such as two pipes filling a tank together, and mixture problems
Common mistake
Skipping the check and keeping an extraneous root.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Rational Equations?
Multiply by the LCD to clear denominators; check extraneous solutions. It is a topic at the advanced level in Algebra.
Where is Rational Equations used in real life?
Real-world examples: Solving combined work problems, such as two pipes filling a tank together, and mixture problems.
What should I learn before Rational Equations?
Make sure you are comfortable with Rational Expression Operations, the previous topic in Algebra.
What is a common mistake with Rational Equations?
Skipping the check and keeping an extraneous root.
How can I practise Rational Equations?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Rational Equations?
Move on to Rational Functions and Asymptotes next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.