Solve for x |4x-8|<0

$| 4 x - 8 | < 0$

Step 1

Write $| 4 x - 8 | < 0$ as a piecewise.

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

$4 x - 8 \geq 0$

Solve the inequality.

Add $8$ to both sides of the inequality.

$4 x \geq 8$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $8$ by $4$ .

$x \geq 2$

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

$4 x - 8 < 0$

Solve the inequality.

Add $8$ to both sides of the inequality.

$4 x < 8$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $8$ by $4$ .

$x < 2$

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

$- (4 x - 8) < 0$

Write as a piecewise.

${\begin{cases} 4 x - 8 < 0 & x \geq 2 \\ - (4 x - 8) < 0 & x < 2 \end{cases}$

Simplify $- (4 x - 8) < 0$ .

Apply the distributive property.

${\begin{cases} 4 x - 8 < 0 & x \geq 2 \\ - (4 x) - - 8 < 0 & x < 2 \end{cases}$

Multiply $4$ by $- 1$ .

${\begin{cases} 4 x - 8 < 0 & x \geq 2 \\ - 4 x - - 8 < 0 & x < 2 \end{cases}$

Multiply $- 1$ by $- 8$ .

${\begin{cases} 4 x - 8 < 0 & x \geq 2 \\ - 4 x + 8 < 0 & x < 2 \end{cases}$

Step 2

Solve $4 x - 8 < 0$ when $x \geq 2$ .

Solve $4 x - 8 < 0$ for $x$ .

Add $8$ to both sides of the inequality.

$4 x < 8$

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

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

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $8$ by $4$ .

$x < 2$

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

No solution

Step 3

Solve $- 4 x + 8 < 0$ when $x < 2$ .

Solve $- 4 x + 8 < 0$ for $x$ .

Subtract $8$ from both sides of the inequality.

$- 4 x < - 8$

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

Divide each term in $- 4 x < - 8$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} > \frac{- 8}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} > \frac{- 8}{- 4}$

Divide $x$ by $1$ .

$x > \frac{- 8}{- 4}$

Simplify the right side.

Divide $- 8$ by $- 4$ .

$x > 2$

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

No solution

Step 4

Find the union of the solutions.

No solution

Solve for x |4x-8|<0

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