Question

Solve and check:

1
x + 3
=
x + 10
x − 2
From least to greatest, the solutions are x =
and x =
.

Answer 1

Interpret the equation as 1/(x+3) = (x+10)/(x−2).

Cross-multiply:
x − 2 = (x + 10)(x + 3) = x^2 + 13x + 30.

Bring all terms to one side:
0 = x^2 + 13x + 30 − x + 2 = x^2 + 12x + 32.

Factor or use quadratic formula:
x^2 + 12x + 32 = (x + 8)(x + 4) = 0,
so x = −8 or x = −4.

Check (denominators x ≠ −3, 2 — both roots are allowed):
x = −8: 1/(−8+3)=1/(−5)=−1/5, (−8+10)/(−8−2)=2/(−10)=−1/5 OK.
x = −4: 1/(−4+3)=1/(−1)=−1, (−4+10)/(−4−2)=6/(−6)=−1 OK.

From least to greatest, x = −8 and x = −4.