If x is a real number that satisfies the absolute value equation |2x - 7| = 3x - 11 what i… — GMAT

x

is a real number that satisfies the absolute value equation

|2x - 7| = 3x - 11

what is the value of

x^2 - 3x

The value of

x^2 - 3x

4

Solving the absolute value equation

|2x - 7| = 3x - 11

yields two potential roots:

x = 4

and

x = 3.6

. Checking both in the original equation shows that

x = 3.6

makes the right-hand side negative (

-0.2

), making it an extraneous solution. The only valid solution is

x = 4

. Substituting

x = 4

into

x^2 - 3x

yields

4^2 - 3(4) = 16 - 12 = 4

Set up equations based on the definition of absolute value

Case 1:

2x - 7 = 3x - 11

; Case 2:

2x - 7 = -(3x - 11)

An absolute value equation

|A| = B

splits into

A = B

A = -B

, with the requirement that

B \ge 0

Solve Case 1:

2x - 7 = 3x - 11

3x - 2x = 11 - 7 \implies x = 4

Isolate the variable

x

by algebraic rearrangement.

Solve Case 2:

2x - 7 = -3x + 11

5x = 18 \implies x = \frac{18}{5} = 3.6

Isolate

x

for the negative case.

Check both potential solutions in the original equation

|2x - 7| = 3x - 11

For

x = 4

|2(4) - 7| = |1| = 1

and

3(4) - 11 = 1

(Valid).
For

x = 3.6

|2(3.6) - 7| = |0.2| = 0.2

but

3(3.6) - 11 = -0.2

(Extraneous).

The right side

3x - 11

must be non-negative. Since

3(3.6) - 11 = -0.2 < 0

x = 3.6

is extraneous.

Evaluate the target expression

x^2 - 3x

using the valid root

x = 4

4^2 - 3(4) = 16 - 12 = 4

Substitute the single valid root into the requested expression.

Absolute Value Linear Equations and Extraneous Solution Checking

Question