Solve for x |3x-8|>7

$| 3 x - 8 | > 7$

Step 1

Write $| 3 x - 8 | > 7$ as a piecewise.

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

$3 x - 8 \geq 0$

Solve the inequality.

Add $8$ to both sides of the inequality.

$3 x \geq 8$

Divide each term in $3 x \geq 8$ by $3$ and simplify.

Divide each term in $3 x \geq 8$ by $3$ .

$\frac{3 x}{3} \geq \frac{8}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{8}{3}$

Divide $x$ by $1$ .

$x \geq \frac{8}{3}$

In the piece where $3 x - 8$ is non-negative, remove the absolute value.

$3 x - 8 > 7$

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

$3 x - 8 < 0$

Solve the inequality.

Add $8$ to both sides of the inequality.

$3 x < 8$

Divide each term in $3 x < 8$ by $3$ and simplify.

Divide each term in $3 x < 8$ by $3$ .

$\frac{3 x}{3} < \frac{8}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} < \frac{8}{3}$

Divide $x$ by $1$ .

$x < \frac{8}{3}$

In the piece where $3 x - 8$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 x - 8) > 7$

Write as a piecewise.

${\begin{cases} 3 x - 8 > 7 & x \geq \frac{8}{3} \\ - (3 x - 8) > 7 & x < \frac{8}{3} \end{cases}$

Simplify $- (3 x - 8) > 7$ .

Apply the distributive property.

${\begin{cases} 3 x - 8 > 7 & x \geq \frac{8}{3} \\ - (3 x) - - 8 > 7 & x < \frac{8}{3} \end{cases}$

Multiply $3$ by $- 1$ .

${\begin{cases} 3 x - 8 > 7 & x \geq \frac{8}{3} \\ - 3 x - - 8 > 7 & x < \frac{8}{3} \end{cases}$

Multiply $- 1$ by $- 8$ .

${\begin{cases} 3 x - 8 > 7 & x \geq \frac{8}{3} \\ - 3 x + 8 > 7 & x < \frac{8}{3} \end{cases}$

Step 2

Solve $3 x - 8 > 7$ for $x$ .

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

Add $8$ to both sides of the inequality.

$3 x > 7 + 8$

Add $7$ and $8$ .

$3 x > 15$

Divide each term in $3 x > 15$ by $3$ and simplify.

Divide each term in $3 x > 15$ by $3$ .

$\frac{3 x}{3} > \frac{15}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} > \frac{15}{3}$

Divide $x$ by $1$ .

$x > \frac{15}{3}$

Simplify the right side.

Divide $15$ by $3$ .

$x > 5$

Step 3

Solve $- 3 x + 8 > 7$ for $x$ .

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

Subtract $8$ from both sides of the inequality.

$- 3 x > 7 - 8$

Subtract $8$ from $7$ .

$- 3 x > - 1$

Divide each term in $- 3 x > - 1$ by $- 3$ and simplify.

Divide each term in $- 3 x > - 1$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} < \frac{- 1}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} < \frac{- 1}{- 3}$

Divide $x$ by $1$ .

$x < \frac{- 1}{- 3}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x < \frac{1}{3}$

Step 4

Find the union of the solutions.

$x < \frac{1}{3}$ or $x > 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{1}{3} or x > 5$

Interval Notation:

$(- \infty, \frac{1}{3}) \cup (5, \infty)$

Step 6

Solve for x |3x-8|>7

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