What is the sum of all integer solutions to the equation |2x - 8| = 14? — ACT

What is the sum of all integer solutions to the equation $|2x - 8| = 14$ ?

The sum of all integer solutions is 8.

To solve

|2x - 8| = 14

, consider both possible cases:

2x - 8 = 14

and

2x - 8 = -14

. The first equation gives

2x = 22

, so

x = 11

. The second equation gives

2x = -6

, so

x = -3

. Adding these two solutions yields

11 + (-3) = 8

Set up the two linear equations represented by the absolute value equation.

2x - 8 = 14

2x - 8 = -14

The absolute value expression

|x| = k

for

k > 0

splits into

x = k

and

x = -k

Solve the first equation for

x

2x = 22 \implies x = 11

Add 8 to both sides and divide by 2.

Solve the second equation for

x

2x = -6 \implies x = -3

Add 8 to both sides and divide by 2.

Calculate the sum of both integer solutions.

11 + (-3) = 8

The problem asks for the sum of all integer solutions.

Solving Absolute Value Equations with Integers

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