What is the real solution to the equation |x - 5| = 2x + 1? — GMAT

What is the real solution to the equation $|x - 5| = 2x + 1$ ?

x = \frac{4}{3}

The correct answer is

x = \frac{4}{3}

. Setting

x - 5 = -(2x + 1)

gives

x - 5 = -2x - 1

, which solves to

3x = 4

x = \frac{4}{3}

. Plugging

x = \frac{4}{3}

into both sides of the original equation yields

\frac{11}{3} = \frac{11}{3}

, satisfying the equation.

Set up the two cases for the absolute value equation

Case 1:

x - 5 = 2x + 1

; Case 2:

x - 5 = -(2x + 1)

By definition,

|A| = B

implies

A = B

A = -B

, provided

B \geq 0

Solve Case 1 algebraically

x - 5 = 2x + 1 \implies x = -6

Subtracting

x

and

1

from both sides isolates

x

Solve Case 2 algebraically

x - 5 = -2x - 1 \implies 3x = 4 \implies x = \frac{4}{3}

Distributing the negative sign and adding

2x

and

5

to both sides isolates

x

Check for extraneous solutions in the original equation

|x - 5| = 2x + 1

For

x = -6

|-6 - 5| = |-11| = 11

, but

2(-6) + 1 = -11 \neq 11

(extraneous). For

x = \frac{4}{3}

|\frac{4}{3} - 5| = |-\frac{11}{3}| = \frac{11}{3}

, and

2(\frac{4}{3}) + 1 = \frac{11}{3}

(valid).

An absolute value expression cannot equal a negative number.

Solving absolute value equations with a variable on the right-hand side requires checking for extraneous solutions.

Question