Pre-Algebra Examples

$| 4 x - 8 | < 8$

Step 1

Write $| 4 x - 8 | < 8$ as a piecewise.

Tap for more steps...

Step 1.1

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

$4 x - 8 \geq 0$

Step 1.2

Solve the inequality.

Tap for more steps...

Step 1.2.1

Add $8$ to both sides of the inequality.

$4 x \geq 8$

Step 1.2.2

Divide each term in $4 x \geq 8$ by $4$ and simplify.

Tap for more steps...

Step 1.2.2.1

Divide each term in $4 x \geq 8$ by $4$ .

$\frac{4 x}{4} \geq \frac{8}{4}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{8}{4}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{8}{4}$

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

Divide $8$ by $4$ .

$x \geq 2$

Step 1.3

In the piece where $4 x - 8$ is non-negative, remove the absolute value.

$4 x - 8 < 8$

Step 1.4

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

$4 x - 8 < 0$

Step 1.5

Solve the inequality.

Tap for more steps...

Step 1.5.1

Add $8$ to both sides of the inequality.

$4 x < 8$

Step 1.5.2

Divide each term in $4 x < 8$ by $4$ and simplify.

Tap for more steps...

Step 1.5.2.1

Divide each term in $4 x < 8$ by $4$ .

$\frac{4 x}{4} < \frac{8}{4}$

Step 1.5.2.2

Simplify the left side.

Tap for more steps...

Step 1.5.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} < \frac{8}{4}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{8}{4}$

Step 1.5.2.3

Simplify the right side.

Tap for more steps...

Step 1.5.2.3.1

Divide $8$ by $4$ .

$x < 2$

Step 1.6

In the piece where $4 x - 8$ is negative, remove the absolute value and multiply by $- 1$ .

$- (4 x - 8) < 8$

Step 1.7

Write as a piecewise.

${\begin{cases} 4 x - 8 < 8 & x \geq 2 \\ - (4 x - 8) < 8 & x < 2 \end{cases}$

Step 1.8

Simplify $- (4 x - 8) < 8$ .

Tap for more steps...

Step 1.8.1

Apply the distributive property.

${\begin{cases} 4 x - 8 < 8 & x \geq 2 \\ - (4 x) - - 8 < 8 & x < 2 \end{cases}$

Step 1.8.2

Multiply $4$ by $- 1$ .

${\begin{cases} 4 x - 8 < 8 & x \geq 2 \\ - 4 x - - 8 < 8 & x < 2 \end{cases}$

Step 1.8.3

Multiply $- 1$ by $- 8$ .

${\begin{cases} 4 x - 8 < 8 & x \geq 2 \\ - 4 x + 8 < 8 & x < 2 \end{cases}$

Step 2

Solve $4 x - 8 < 8$ when $x \geq 2$ .

Tap for more steps...

Step 2.1

Solve $4 x - 8 < 8$ for $x$ .

Tap for more steps...

Step 2.1.1

Move all terms not containing $x$ to the right side of the inequality.

Tap for more steps...

Step 2.1.1.1

Add $8$ to both sides of the inequality.

$4 x < 8 + 8$

Step 2.1.1.2

Add $8$ and $8$ .

$4 x < 16$

Step 2.1.2

Divide each term in $4 x < 16$ by $4$ and simplify.

Tap for more steps...

Step 2.1.2.1

Divide each term in $4 x < 16$ by $4$ .

$\frac{4 x}{4} < \frac{16}{4}$

Step 2.1.2.2

Simplify the left side.

Tap for more steps...

Step 2.1.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} < \frac{16}{4}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{16}{4}$

Step 2.1.2.3

Simplify the right side.

Tap for more steps...

Step 2.1.2.3.1

Divide $16$ by $4$ .

$x < 4$

Step 2.2

Find the intersection of $x < 4$ and $x \geq 2$ .

$2 \leq x < 4$

Step 3

Solve $- 4 x + 8 < 8$ when $x < 2$ .

Tap for more steps...

Step 3.1

Solve $- 4 x + 8 < 8$ for $x$ .

Tap for more steps...

Step 3.1.1

Move all terms not containing $x$ to the right side of the inequality.

Tap for more steps...

Step 3.1.1.1

Subtract $8$ from both sides of the inequality.

$- 4 x < 8 - 8$

Step 3.1.1.2

Subtract $8$ from $8$ .

$- 4 x < 0$

Step 3.1.2

Divide each term in $- 4 x < 0$ by $- 4$ and simplify.

Tap for more steps...

Step 3.1.2.1

Divide each term in $- 4 x < 0$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} > \frac{0}{- 4}$

Step 3.1.2.2

Simplify the left side.

Tap for more steps...

Step 3.1.2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} > \frac{0}{- 4}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{0}{- 4}$

Step 3.1.2.3

Simplify the right side.

Tap for more steps...

Step 3.1.2.3.1

Divide $0$ by $- 4$ .

$x > 0$

Step 3.2

Find the intersection of $x > 0$ and $x < 2$ .

$0 < x < 2$

Step 4

Find the union of the solutions.

$0 < x < 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 4$

Interval Notation:

$(0, 4)$

Step 6