If x is a real number satisfying the equation |2x - 3| = 5x - 12, what is the sum of all v… — GMAT

If $x$ is a real number satisfying the equation $|2x - 3| = 5x - 12$ , what is the sum of all valid real solutions for $x$ ?

$3$ Answer
B
$\frac{15}{7}$
C
$\frac{36}{7}$
D
$-3$
E
$\frac{45}{7}$

Answer

The sum of all valid real solutions is

3

The option specifying

3

correctly identifies

x = 3

as the only valid solution after discarding the extraneous root

x = \frac{15}{7}

, which produces a negative right-hand side in the original equation.

Step-by-Step Solution

Set up the two algebraic cases for the absolute value equation

|2x - 3| = 5x - 12

Case 1:

2x - 3 = 5x - 12

Case 2:

2x - 3 = -(5x - 12) = -5x + 12

By definition,

|a| = b

implies

a = b

a = -b

, provided

b \ge 0

Solve Case 1 for

x

2x - 3 = 5x - 12 \implies 3x = 9 \implies x = 3

Isolate the variable

x

algebraically.

Solve Case 2 for

x

2x - 3 = -5x + 12 \implies 7x = 15 \implies x = \frac{15}{7}

Isolate the variable

x

algebraically.

Check candidate solutions for extraneous roots in the original equation.

For

x = 3

|2(3) - 3| = |3| = 3

and

5(3) - 12 = 3

. Valid.
For

x = \frac{15}{7}

|2(\frac{15}{7}) - 3| = |\frac{9}{7}| = \frac{9}{7}

, but

5(\frac{15}{7}) - 12 = -\frac{9}{7}

. Since

\frac{9}{7} \neq -\frac{9}{7}

x = \frac{15}{7}

is extraneous.

The output of an absolute value expression cannot be negative, so any candidate root making the right side negative must be discarded.

Calculate the sum of all valid real solutions.

Sum =

3

There is only one valid solution,

x = 3

Key Concept

Solving absolute value equations with a variable expression on the right-hand side requires checking candidate solutions to eliminate extraneous roots.

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