If |x - 5| = 2x - 4, what is the value of x? — GMAT

If $|x - 5| = 2x - 4$ , what is the value of $x$ ?

The value of

x

is 3.

Solving the absolute value equation yields two candidate values,

x = 3

and

x = -1

. Substituting

x = 3

into the original equation gives

|3 - 5| = 2

and

2(3) - 4 = 2

, which makes the statement true. Therefore,

x = 3

is the only valid solution.

Set up the two cases for the absolute value equation

|x - 5| = 2x - 4

Case 1:

x - 5 = 2x - 4

; Case 2:

x - 5 = -(2x - 4)

By definition,

|a| = b

implies

a = b

a = -b

when

b \ge 0

Solve Case 1:

x - 5 = 2x - 4

x = -1

Subtract

x

and add

4

to both sides.

Solve Case 2:

x - 5 = -2x + 4

3x = 9 \implies x = 3

Add

2x

and add

5

to both sides.

Test potential solutions in the original equation to eliminate extraneous roots.

For

x = -1

|-1 - 5| = 6 \neq 2(-1) - 4 = -6

(Invalid). For

x = 3

|3 - 5| = 2 = 2(3) - 4 = 2

(Valid).

The absolute value of a real number cannot be negative, so the right-hand side

2x - 4

must be non-negative.

Absolute Value Equations and Extraneous Solutions

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