If y is a real number such that |3 - \frac{1}{2}y| = 5, what is the sum of all possible va… — ACT

If $y$ is a real number such that $|3 - \frac{1}{2}y| = 5$ , what is the sum of all possible values of $y$ ?

Solving the equation

|3 - \frac{1}{2}y| = 5

requires setting up two cases:

3 - \frac{1}{2}y = 5

and

3 - \frac{1}{2}y = -5

. Solving the first case gives

y = -4

, and solving the second case gives

y = 16

. The sum of these two solutions is

-4 + 16 = 12

Set up the two equations represented by the absolute value expression.

3 - \frac{1}{2}y = 5

and

3 - \frac{1}{2}y = -5

An absolute value equation

|u| = C

(where

C \ge 0

) splits into two cases:

u = C

and

u = -C

Solve the first equation

3 - \frac{1}{2}y = 5

y = -4

Subtracting 3 from both sides yields

-\frac{1}{2}y = 2

. Multiplying both sides by

-2

isolates

y

Solve the second equation

3 - \frac{1}{2}y = -5

y = 16

Subtracting 3 from both sides yields

-\frac{1}{2}y = -8

. Multiplying both sides by

-2

isolates

y

Calculate the sum of all possible values of

y

12

Add the two solutions together:

-4 + 16 = 12

Absolute Value Equations

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