An absolute value measures an expression's distance from a midpoint on a number line.

∣ - 6 x ∣ = 24

This equation means that the distance is

24,

either in the

positive

direction or the

negative

direction.

∣ - 6 x ∣ = 24 \Rightarrow - 6 x = - 24 - 6 x = - 24

To find the solutions to the absolute value equation, we need to solve both of these cases for

x .

∣ - 6 x ∣ = 24

$- 6 x \geq 0 : - 6 x = 24 - 6 x < 0 : - 6 x = - 24 (I) (II)$

- 6 x = 24 - 6 x = - 24 (I) (II)

$(I), (II):$ $LHS / - 6 = RHS / - 6$

x = - 4 x = 4

Checking Our Answers

When solving an absolute value equation, it is important to check for extraneous solutions. We can check our answers by substituting them back into the original equation. Let's start with

4

.

∣ - 6 x ∣ = 24

Substitute

$x = 4$

∣ - 6 (4) ∣ = ? 24

Multiply

∣ - 24 ∣ = ? 24

AbsNeg

$∣ - 24 ∣ = 24$

24 = 24

We will check

- 4

in the same way.

∣ - 6 x ∣ = 24

Substitute

$x = - 4$

∣ - 6 (- 4) ∣ = ? 24

MultNegNegOnePar

$- a (- b) = a \cdot b$

∣ 24 ∣ = ? 24

AbsPos

$∣ 24 ∣ = 24$

24 = 24

We see that both

4

and

- 4

satisfy the original equation.

Exercise