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Equations with x in the Denominator

Here you’ll learn how to get rid of a denominator containing variables. The good news is that the approach is the same as with getting rid of a denominator with numbers! To find the common denominator, you multiply all the different factors once. Here’s two examples:

Example 1

Solve the equation 2 x = 3

2 x = 3 x × 2 x = 3 × x 2 = 3x 2 3 = 3x 3 2 3 = x x = 2 3

Example 2

Solve the equation x + 1 x + 2 = 3 x + 1 + 3 for x

The common denominator is x(x + 1). Multiply both sides of the equation with the common denominator:

= x(x + 1) × (x + 1 x + 2) = x(x + 1) × ( 3 x + 1 + 3) .

x(x + 1) × (x + 1 x + 2) = x(x + 1) × ( 3 x + 1 + 3) .

This expands to
= x(x + 1) ×x + 1 x + x(x + 1) × 2 = x(x + 1) × 3 x + 1 + x(x + 1) × 3.

x(x + 1) ×x + 1 x + x(x + 1) × 2 = x(x + 1) × 3 x + 1 + x(x + 1) × 3.

Here you can cancel some factors:
= x(x + 1)x + 1 x + x(x + 1)2 = x(x + 1) 3 x + 1 + x(x + 1)3.

x(x + 1)x + 1 x + x(x + 1)2 = x(x + 1) 3 x + 1 + x(x + 1)3.

Now the expression simplifies to
= x2 + 2x + 1 + 2x2 + 2x = 3x + 3x2 + 3x.

x2 + 2x + 1 + 2x2 + 2x = 3x + 3x2 + 3x.

Isolate all the variables on one side and the constants on the other:
1 = x2 + 2x2 3x2 + 2x + 2x 3x 3x.

x2 + 2x2 3x2 + 2x + 2x 3x 3x = 1.

This simplifies to
2x = 1.

Get rid of the number in front of x:

2x 2 = 1 2 = 1 2.

Therefore, x = 1 2.

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