If x is a real number satisfying the equation |x - 4| = 2x - 1, what is the value of |x - … — GMAT

If $x$ is a real number satisfying the equation $|x - 4| = 2x - 1$ , what is the value of $|x - 5|$ ?

\frac{10}{3}

Solving

|x - 4| = 2x - 1

requires

2x - 1 \geq 0

, or

x \geq \frac{1}{2}

. Splitting into cases gives

x - 4 = 2x - 1 \implies x = -3

(invalid because

-3 < \frac{1}{2}

) and

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

(valid because

\frac{5}{3} \geq \frac{1}{2}

). Substituting

x = \frac{5}{3}

into

|x - 5|

yields

|\frac{5}{3} - 5| = |-\frac{10}{3}| = \frac{10}{3}

Determine the non-negativity constraint for the absolute value equation

Since absolute values are non-negative,

2x - 1 \geq 0 \implies x \geq \frac{1}{2}

The right-hand side of

|x - 4| = 2x - 1

must be greater than or equal to 0.

Split into linear cases and solve for candidate values of x

Case 1:

x - 4 = 2x - 1 \implies x = -3

.
Case 2:

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

An absolute value equation

|A| = B

splits into

A = B

A = -B

Eliminate extraneous solutions

x = -3

fails the condition

x \geq \frac{1}{2}

(since

2(-3) - 1 = -7 < 0

x = \frac{5}{3}

satisfies

x \geq \frac{1}{2}

(since

2(\frac{5}{3}) - 1 = \frac{7}{3} > 0

). Thus,

x = \frac{5}{3}

is the sole valid solution.

Candidate solutions must satisfy the original non-negativity constraint.

Substitute the valid solution into

|x - 5|

|\frac{5}{3} - 5| = |-\frac{10}{3}| = \frac{10}{3}

Calculate the value of the requested expression using

x = \frac{5}{3}

Absolute Value Equations and Extraneous Solutions

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