4. Solve for x and check.

 

This is a rational equation so we run through the steps for solving rational equations. First, clear denominators. Multiply both sides by a common denominator. In this case the smallest common denominator is x(x-2).

 

 

Since the left side has two terms, we need to multiply them both.

 

 

Assemble the surviving factors

 

 

Remove parentheses.

 

 

When we combine like terms we see that we have a quadratic. Transpose all the terms to the left leaving a 0 on the right.

 

 

Now that we have a zero on one side we see if the left side factors. It does.

 

 

Set the factors equal to zero.

 

 

Check:

Let x = 4/3

 

Copy down the original equation

 

 

except that wherever you see an x, copy a 4/3 in parentheses.

 

 

The right simplifies when the 3's cancel, but we have more work to do on the left. In the bottom of the fraction, we have a subtraction of fraction problem, so we need common denominators.

 

 

This changes the 2 into a 6/3

 

 

or

 

 

We are now ready to invert and multiply in the compound fraction in the first term on the right.

 

 

We need common denominators on the left.

 

 

which checks.

 

Let x = -1.

 

Copy down the original equation

 

 

except that wherever you see an x, copy down a -1 in parentheses.

 

 

Do the arithmetic on the bottoms.

 

 

Find common denominators on the left.

 

 

which checks.

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