Solve the Absolute Value Inequality for x |x|<8

$| x | < 8$

Step 1

Write $| x | < 8$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

In the piece where $x$ is non-negative, remove the absolute value.

$x < 8$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x < 8$

Write as a piecewise.

${\begin{cases} x < 8 & x \geq 0 \\ - x < 8 & x < 0 \end{cases}$

Step 2

Find the intersection of $x < 8$ and $x \geq 0$ .

$0 \leq x < 8$

Step 3

Solve $- x < 8$ when $x < 0$ .

Divide each term in $- x < 8$ by $- 1$ and simplify.

Divide each term in $- x < 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{8}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{8}{- 1}$

Divide $x$ by $1$ .

$x > \frac{8}{- 1}$

Simplify the right side.

Divide $8$ by $- 1$ .

$x > - 8$

Find the intersection of $x > - 8$ and $x < 0$ .

$- 8 < x < 0$

Step 4

Find the union of the solutions.

$- 8 < x < 8$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 8 < x < 8$

Interval Notation:

$(- 8, 8)$

Step 6

Solve the Absolute Value Inequality for x |x|<8

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