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Equations with x in the Denominator


Having learned how to use common denominators to get rid of fractions in equations, you might be wondering how to isolate a variable that is in the denominator. This is a great question!

The good news is that the approach is the same as with getting rid of a denominator that contains only 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 by 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

Now, 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

Finally, get rid of the number in front of x:

2x 2 = 1 2 = 1 2

Therefore, x = 1 2.

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