# Rational Equation

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

## Example3/x + x/(x - 1) = 1/[x(x - 1)]

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 = +x^{2}

Move 1 to the left side.

Then x^{2} + 3x - 4 = 0.

Factor the left side

x^{2} + 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

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.