If x is a real number that satisfies the equation |2x - 1| = 3x + 11, what is the value of… — GMAT

If $x$ is a real number that satisfies the equation $|2x - 1| = 3x + 11$ , what is the value of $x^2 + 2x$ ?

The value of

x^2 + 2x

0

Solving the equation

|2x - 1| = 3x + 11

produces two algebraic candidates:

x = -12

and

x = -2

. Substituting

x = -12

into the right side yields

3(-12) + 11 = -25

. Since absolute value expressions cannot be negative,

x = -12

is an extraneous solution. Substituting

x = -2

yields

|-5| = 5

, which is valid. Evaluating

x^2 + 2x

x = -2

gives

(-2)^2 + 2(-2) = 0

Set up the two linear cases for the absolute value equation

|2x - 1| = 3x + 11

Case 1:

2x - 1 = 3x + 11

; Case 2:

2x - 1 = -(3x + 11)

By definition,

|u| = c

implies

u = c

u = -c

(provided

c \geq 0

Solve each linear equation for candidate values of

x

From Case 1:

x = -12

. From Case 2:

2x - 1 = -3x - 11 \implies 5x = -10 \implies x = -2

Isolate

x

algebraically in both equations.

Check candidate solutions in the original equation to eliminate extraneous roots.

For

x = -12

|2(-12) - 1| = |-25| = 25

, but

3(-12) + 11 = -25 \neq 25

(extraneous). For

x = -2

|2(-2) - 1| = |-5| = 5

, and

3(-2) + 11 = 5

(valid solution).

An absolute value cannot equal a negative number; substituting back is mandatory.

Evaluate the target expression

x^2 + 2x

using the valid root

x = -2

(-2)^2 + 2(-2) = 4 - 4 = 0

Substitute the verified real solution into the given expression.

Solving Absolute Value Linear Equations and Checking for Extraneous Solutions

Estimated Time:1m 30s

Question