What is a rational equation?
A rational equation is an equation with at least one fraction that has a variable in the denominator, such as 3/(x + 1) = 2/x. A fraction like x/4 doesn't count, because its denominator is just a number.
The variable in the bottom is what makes these problems different. Some values of x make a denominator zero, and those can never be answers. Keeping track of them is half the job.
Type the equation with parentheses around each denominator, for example (x+1)/(x-3) = 2 or 1/x + 1/(x+1) = 1/2.
What are the steps for solving rational equations?
Find the excluded values, multiply by the LCD, solve, and check. Solving rational equations by hand follows the same four steps the calculator shows.
- Factor each denominator and write down the values of x that make any of them zero.
- Find the least common denominator (LCD) of all the fractions.
- Multiply every term on both sides by the LCD. The fractions disappear.
- Solve the equation that is left. It is usually linear or quadratic.
- Cross out any answer that matches an excluded value, then check the rest in the original equation.
What is an extraneous solution in a rational equation?
An extraneous solution is a number you get from the cleared equation that doesn't work in the original, because it makes a denominator zero. Multiplying by the LCD can create these, since the LCD itself is zero at the excluded values.
Look at x/(x − 2) = 2/(x − 2). The excluded value is x = 2. Multiplying both sides by x − 2 gives x = 2, which is the one value that isn't allowed. So the calculator rejects it and reports No solution.
This is why the excluded values come first. If you skip that step, x = 2 looks like a perfectly good answer.
How do you find the LCD of polynomial denominators?
Factor each denominator, then take every different factor the greatest number of times it appears in any one denominator. That's the same rule as the least common multiple of whole numbers, only with factors like x − 1 in place of primes.
| Denominators | LCD | Why |
|---|---|---|
| x and x + 1 | x(x + 1) | No shared factors, so multiply them |
| x, x + 1 and 2 | 2x(x + 1) | Each factor appears once |
| x − 1, x + 1 and x² − 1 | (x + 1)(x − 1) | x² − 1 already contains both factors |
| x and x² | x² | Use the highest power of x |
The third row is a common textbook problem: 2/(x − 1) − 1/(x + 1) = 1/(x² − 1). Since x² − 1 = (x + 1)(x − 1), that product is already the LCD. Multiplying through gives 2(x + 1) − (x − 1) = 1, which simplifies to x + 3 = 1, so x = −2. It isn't 1 or −1, so it stays. Both sides come out to 1/3 when you check it.
When can you cross multiply a rational equation?
You can cross multiply when the equation is a proportion: exactly one fraction on each side and nothing else. It's a shortcut for multiplying by the LCD.
For 3/(x + 1) = 2/x, cross multiplying gives 3x = 2(x + 1), so 3x = 2x + 2 and x = 2. That isn't 0 or −1, and the check works: 3/3 = 1 and 2/2 = 1.
Cross multiplying can also lead to a quadratic. With x/3 = 4/(x + 1), you get x(x + 1) = 12, then x² + x − 12 = 0. That factors to (x + 4)(x − 3) = 0, so x = −4 or x = 3. Neither is the excluded value −1, so both answers stand.
Once a side has two terms, such as 1/x + 1/(x + 1) = 1/2, cross multiplying no longer applies. Use the LCD method instead.
What if clearing the fractions gives a quadratic?
Solve it like any quadratic, then check both roots against the excluded values. The calculator uses the quadratic formula when the result doesn't factor nicely.
Neither root is 0 or −1, so both are kept. If you want to see the quadratic formula step on its own, the quadratic equation solver walks through it.
What does "all real numbers except" mean?
It means the equation is true for every x you're allowed to use. Only the excluded values are left out.
Try x/(x + 1) = 1 − 1/(x + 1). Multiplying by x + 1 gives x = (x + 1) − 1, which is x = x. That's true for any number, but x = −1 was never allowed. The calculator's answer is All real numbers except x = −1.
What are common mistakes when you solve rational equations?
Most lost marks come from three habits: skipping the excluded values, multiplying only some terms by the LCD, and dropping parentheses. In 2/(x − 1) − 1/(x + 1), the second numerator after clearing is −(x − 1), which is −x + 1, not −x − 1.
Another slip is cross multiplying when a side has two fractions. And don't confuse a rational equation with a rational expression. If there's no equals sign, you simplify instead, which is what the rational expressions calculator does.
One limit to know: this rational equations solver handles equations only. Rational inequalities such as 1/x < 2 aren't supported.