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.

x2 - x + 4 = 4x

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

x2 - x - 4x + 4 = 0

x2 - 5x + 4 = 0

Factor

(x - 1)(x - 4) = 0

Set the factors equal to 0.

x - 1 = 0     x - 4 = 0

x = 1       x = 4

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.

4/0 would be how long it would take to travel 4 km. if you didn't move. If you didn't move, it would take forever to travel 4 km. 4/0 - 1/4 would be how long it would take to travel 4 km. if you started keeping time 1/4 hour after you started. It would still take forever, so the solution does make some kind of sense. However, many authors find this to be just too wierd, and will adopt the convention that the solution does not check if we get zero denominators in the check.