6. Solve for x and check.

 

This is a rational equation, so we need to run through the steps for solving a rational equation. First, clear denominators. Multiply both sides by a common denominator. In this case the smallest common denominator for all the denominators is x(x-4). Multiply both sides by x(x-4)

 

 

The big parentheses are there because we were adding the two fractions before we ever thought of multiplying the left side by anything, but they tell us to multiply both fractions on the left by x(x-4) when we remove them.

 

 

Assemble the surviving factors.

 

 

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

 

 

Factor

 

 

Set the factors equal to 0.

 

 

Check: x = 1

 

Copy down the original equation,

 

 

except substitute a 1 in for x in the equation.

 

 

Subtract in the bottoms.

 

i.e.,

 

 

which checks.

 

Check: x = 4

 

Copy down the original equation,

 

 

except substitute a 4 in for x in the equation.

 

 

In this case, when we do the subtraction in the bottoms we get

 

 

which has zero denominators. If a solution gives us zero denominators when we do the check, we consider that the solution does not check.

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