What is the sum of all valid real solutions to the equation |2x + 15| = 1 - 5x? — GMAT

What is the sum of all valid real solutions to the equation $|2x + 15| = 1 - 5x$ ?

The sum of all valid real solutions is

-2

Solving the absolute value equation

|2x + 15| = 1 - 5x

yields two candidate values:

x = -2

and

x = \frac{16}{3}

. Substituting

x = -2

into the right-hand side gives

1 - 5(-2) = 11

, which is non-negative and matches

|2(-2) + 15| = 11

. Substituting

x = \frac{16}{3}

gives

1 - 5\left(\frac{16}{3}\right) = -\frac{77}{3}

, which is negative and therefore invalid. Thus,

x = -2

is the unique valid solution, making the sum

-2

Set up the two cases for the absolute value equation

|2x + 15| = 1 - 5x

Case 1:

2x + 15 = 1 - 5x

; Case 2:

2x + 15 = -(1 - 5x)

By definition,

|u| = v

implies

u = v

u = -v

, provided

v \geq 0

Solve Case 1 for

x

7x = -14 \implies x = -2

Adding

5x

and subtracting

15

from both sides isolates

x

Solve Case 2 for

x

2x + 15 = -1 + 5x \implies 3x = 16 \implies x = \frac{16}{3}

Distributing the negative sign and combining like terms yields

x = \frac{16}{3}

Check candidate solutions against the non-negativity constraint

1 - 5x \geq 0

For

x = -2

1 - 5(-2) = 11 \geq 0

(Valid). For

x = \frac{16}{3}

1 - 5\left(\frac{16}{3}\right) = -\frac{77}{3} < 0

(Extraneous).

An absolute value cannot equal a negative number, so candidate solutions that make the right-hand side negative must be discarded.

Sum all valid real solutions.

The only valid solution is

x = -2

, so the sum is

-2

Extraneous solutions are excluded from the final sum.

Solving absolute value linear equations requires checking candidate solutions against domain constraints to filter out extraneous roots.

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