Convert to Set Notation |5x-8|<13

$| 5 x - 8 | < 13$

Step 1

Solve $- 1 < x < \frac{21}{5}$ .

Write $| 5 x - 8 | < 13$ as a piecewise.

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

$5 x - 8 \geq 0$

Solve the inequality.

Add $8$ to both sides of the inequality.

$5 x \geq 8$

Divide each term in $5 x \geq 8$ by $5$ and simplify.

Divide each term in $5 x \geq 8$ by $5$ .

$\frac{5 x}{5} \geq \frac{8}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{8}{5}$

Divide $x$ by $1$ .

$x \geq \frac{8}{5}$

In the piece where $5 x - 8$ is non-negative, remove the absolute value.

$5 x - 8 < 13$

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

$5 x - 8 < 0$

Solve the inequality.

Add $8$ to both sides of the inequality.

$5 x < 8$

Divide each term in $5 x < 8$ by $5$ and simplify.

Divide each term in $5 x < 8$ by $5$ .

$\frac{5 x}{5} < \frac{8}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} < \frac{8}{5}$

Divide $x$ by $1$ .

$x < \frac{8}{5}$

In the piece where $5 x - 8$ is negative, remove the absolute value and multiply by $- 1$ .

$- (5 x - 8) < 13$

Write as a piecewise.

${\begin{cases} 5 x - 8 < 13 & x \geq \frac{8}{5} \\ - (5 x - 8) < 13 & x < \frac{8}{5} \end{cases}$

Simplify $- (5 x - 8) < 13$ .

Apply the distributive property.

${\begin{cases} 5 x - 8 < 13 & x \geq \frac{8}{5} \\ - (5 x) - - 8 < 13 & x < \frac{8}{5} \end{cases}$

Multiply $5$ by $- 1$ .

${\begin{cases} 5 x - 8 < 13 & x \geq \frac{8}{5} \\ - 5 x - - 8 < 13 & x < \frac{8}{5} \end{cases}$

Multiply $- 1$ by $- 8$ .

${\begin{cases} 5 x - 8 < 13 & x \geq \frac{8}{5} \\ - 5 x + 8 < 13 & x < \frac{8}{5} \end{cases}$

Solve $5 x - 8 < 13$ when $x \geq \frac{8}{5}$ .

Solve $5 x - 8 < 13$ for $x$ .

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

Add $8$ to both sides of the inequality.

$5 x < 13 + 8$

Add $13$ and $8$ .

$5 x < 21$

Divide each term in $5 x < 21$ by $5$ and simplify.

Divide each term in $5 x < 21$ by $5$ .

$\frac{5 x}{5} < \frac{21}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} < \frac{21}{5}$

Divide $x$ by $1$ .

$x < \frac{21}{5}$

Find the intersection of $x < \frac{21}{5}$ and $x \geq \frac{8}{5}$ .

$\frac{8}{5} \leq x < \frac{21}{5}$

Solve $- 5 x + 8 < 13$ when $x < \frac{8}{5}$ .

Solve $- 5 x + 8 < 13$ for $x$ .

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

Subtract $8$ from both sides of the inequality.

$- 5 x < 13 - 8$

Subtract $8$ from $13$ .

$- 5 x < 5$

Divide each term in $- 5 x < 5$ by $- 5$ and simplify.

Divide each term in $- 5 x < 5$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} > \frac{5}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} > \frac{5}{- 5}$

Divide $x$ by $1$ .

$x > \frac{5}{- 5}$

Simplify the right side.

Divide $5$ by $- 5$ .

$x > - 1$

Find the intersection of $x > - 1$ and $x < \frac{8}{5}$ .

$- 1 < x < \frac{8}{5}$

Find the union of the solutions.

$- 1 < x < \frac{21}{5}$

Step 2

Use the inequality $- 1 < x < \frac{21}{5}$ to build the set notation.

${x | - 1 < x < \frac{21}{5}}$

Step 3

Convert to Set Notation |5x-8|<13

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