Rational Equation

How to solve a rational equation: 1 example and its solutions.

Example

Example

Solution

Before solving the equation,
first set (denominator) ≠ 0.

Excluded Value

See 3/x.

The denominator is x.

So x ≠ 0.

See x/(x - 1)

The denominator is (x - 1).

So x - 1 ≠ 0.
Then x ≠ 1.

See 1/[x(x - 1)].

The denominator is x(x - 1).

You just found that
x ≠ 0 and x - 1 ≠ 0.

So x(x - 1) is not 0.

From the denominators,
you found that
x ≠ 0, x ≠ 1.

The x values should satisfy these conditions.

Next, solve the rational equation.

First, find the least common multiple, LCM,
of the denominators.

The denominators are
x, (x - 1), and x(x - 1).

So the LCM is
x(x - 1).

Multiply the LCM x(x - 1)
to both sides.

[3/x]⋅x(x - 1)
= 3(x - 1)

+[x/(x - 1)]⋅x(x - 1)
= +x⋅x

1/[x(x - 1)]⋅x(x - 1)
= 1

Simplify a Rational Expression

3(x - 1) = 3x - 3

+x⋅x = +x2

Move 1 to the left side.

Then x2 + 3x - 4 = 0.

Factor the left side
x2 + 3x - 4.

Find a pair of numbers
whose product is the constant term -4
and whose sum is the coefficient of the middle term +3.

-1⋅4 = -4
-1 + 4 = +3

Then (x - 1)(x + 4) = 0.

Factor a Quadratic Trinomial

Find the zeros.

Quadratic Equation: by Factoring

Case 1) x - 1 = 0
Then x = 1.

This should satisfy
x ≠ 0, x ≠ 1.

But x = 1 doesn't satisfy x ≠ 1.

So x = 1 cannot be the solution.
(= This is the excluded value.)

Case 2) x + 4 = 0
Then x = -4.

This should satisfy
x ≠ 0, x ≠ 1.

x = 4 satisfies both conditions.

So x = 4 can be the solution.

Case 1) No root
Case 2) x = -4

So x = -4.

So x = -4 is the answer.