Question

Ginger and Meredith solved
√
29
−
7
x
=
3
−
x
29
−
7
𝑥
=
3
−
𝑥
.

Ginger claims
x
=
−
5
𝑥
=
−
5
is a solution.
Meredith claims
x
=
4
𝑥
=
4
is a solution.
Which statement is true?

Answer 1

Start with sqrt(29 − 7x) = 3 − x. Because the left side is nonnegative, 3 − x ≥ 0 so x ≤ 3.

Square both sides: 29 − 7x = (3 − x)^2 = x^2 − 6x + 9.
Rearrange: x^2 + x − 20 = 0 → (x + 5)(x − 4) = 0, so x = −5 or x = 4.

Check against the original equation (or the domain x ≤ 3): x = −5 gives sqrt(29 − 7(−5)) = sqrt(64) = 8 and 3 − (−5) = 8, so it works. x = 4 gives sqrt(1) = 1 but 3 − 4 = −1, so it does not satisfy the original equation.

Therefore Ginger (x = −5) is correct and Meredith (x = 4) is not.