Question

Riley makes a mistake in step 2 while doing her homework. What was her mistake?

StartFraction x Over x squared minus 5 x + 6 EndFraction + StartFraction x Over x + 3 EndFraction

Step 1: StartFraction x Over (x minus 2)(x minus 3) EndFraction + StartFraction 3 Over x + 3 EndFraction

Step 2: StartFraction x Over (x minus 2) (x + 3) EndFraction + 3 (x minus 2) Over (x minus 2) (x + 3) EndFraction

Step 3: StartFraction x + 3 x minus 6 Over (x minus 2) (x + 3) EndFraction

Step 4: StartFraction 4 x minus 6 Over (x minus 2) (x + 3) EndFraction

Answer 1

Riley used the wrong common denominator in Step 2 (and so changed factors incorrectly). She replaced (x − 3) by (x + 3) and wrote the sum over (x − 2)(x + 3) instead of using a denominator that contains the original factor (x − 3). Also the second numerator was incorrectly 3 in Step 1 (it should be x), so multiplying to get a common denominator was done incorrectly.

Correctly, starting from
x/((x−2)(x−3)) + x/(x+3),

the common denominator is (x−2)(x−3)(x+3). So
x/((x−2)(x−3)) = x(x+3)/[(x−2)(x−3)(x+3)]
and
x/(x+3) = x(x−2)(x−3)/[(x+3)(x−2)(x−3)].

Thus the sum is
[x(x+3) + x(x−2)(x−3)] / [(x−2)(x−3)(x+3)].