Multiply every term by the least common denominator. The fractions disappear and an ordinary equation remains.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Every term
Missing one term when multiplying through is the most common error, and it silently changes the equation.
Extraneous solutions
Multiplying by an expression that could be zero can introduce a solution that does not satisfy the original.
Why they appear
If the LCD is zero at some value, multiplying by it there is multiplying by zero, which is not a reversible step.
Checking is compulsory
Substitute every answer into the original. Any value making a denominator zero must be rejected.
Sometimes nothing survives
An equation can have every candidate rejected, leaving no solution at all. That is a legitimate answer.
Multiply through by the common denominator
Clearing the fractions turns the equation into a polynomial one, which you already know how to solve. Every term on both sides must be multiplied, including any without a denominator.
Clearing can invent solutions
Multiplying by an expression that could be zero is not reversible, so solutions may appear that make a denominator zero. Every answer must be checked against the original domain.
Check in the original equation
Substituting into the cleared version will not detect the problem, because the cleared version has a larger domain. The check must use the equation as first given.
Discarding is the correct outcome
An equation whose only algebraic solution is excluded has no solution. Reporting that is a complete answer, not a failure to finish.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 0.50)
Step 3: Watch an Example
One step at a time.
Watch Marcus Reject a Solution
Marcus solves x ÷ (x − 2) = 2 ÷ (x − 2).
- Step 1
The common denominator is (x − 2), so he multiplies both sides by it.
Step 4: Your Turn
Practice makes it stick.
The Rejection
Problem 1 of 2
x ÷ (x − 2) = 2 ÷ (x − 2). How many valid solutions?
The Simple Case
Problem 2 of 2
6 ÷ x = 3. What is x?
Clear and Check
1 of 8
12 ÷ x = 4. What is x?
2 of 8
10 ÷ x = 5. What is x?
3 of 8
1 ÷ (x − 3) = 1. What is x?
4 of 8
A candidate makes a denominator zero. Is it valid? 1 yes, 0 no.
5 of 8
8 ÷ (x + 1) = 2. What is x?
6 of 8
Is checking optional for rational equations? 1 yes, 0 no.
7 of 8
Put the solving process in order.
- 1Multiply every term by it.
- 2Solve the resulting equation.
- 3Check each answer in the original and reject any that break it.
- 4Find the least common denominator.
8 of 8
20 ÷ x = 4. What is x?
Step 5: Quick Check
Show what you know.
Question 1 of 2
15 ÷ x = 5. What is x?
Question 2 of 2
Why can extraneous solutions appear?
What You Learned
- Clear denominators by multiplying every term by the LCD.
- That step can introduce extraneous solutions.
- Check every answer in the original equation and reject any that break it.