If x satisfies the equation |8x + 13| = 10x + 5, what is the sum of all valid real solutio… — GMAT

If $x$ satisfies the equation $|8x + 13| = 10x + 5$ , what is the sum of all valid real solutions for $x$ ?

The sum of all valid real solutions for

x

is 4.

The solution

x = 4

is the only value of

x

that satisfies

|8x + 13| = 10x + 5

because evaluating both sides yields

45 = 45

. The alternative algebraic candidate,

x = -1

, produces a right-hand side of

-5

, which is impossible for an absolute value expression. Thus, the sum of all valid real solutions is simply 4.

Split the absolute value equation into two linear case equations

Case 1:

8x + 13 = 10x + 5

; Case 2:

8x + 13 = -(10x + 5)

By definition of absolute value,

|u| = v

implies

u = v

u = -v

, provided

v \ge 0

Solve Case 1 for

x

2x = 8 \implies x = 4

Subtract

8x

and

5

from both sides to isolate

x

Solve Case 2 for

x

8x + 13 = -10x - 5 \implies 18x = -18 \implies x = -1

Expand the right-hand side and combine like terms to isolate

x

Test candidate solutions in the original equation to eliminate extraneous roots

For

x = 4

|8(4)+13| = |45| = 45

and

10(4)+5 = 45

(Valid). For

x = -1

|8(-1)+13| = |5| = 5

and

10(-1)+5 = -5

(Extraneous).

An absolute value output cannot equal a negative number.

Calculate the sum of all valid solutions

The only valid solution is

x = 4

, so the sum is

4

Only valid roots must be summed.

Absolute Value Equations and Checking for Extraneous Solutions

Question