Solve for s |2s-3|-6>-5

$| 2 s - 3 | - 6 > - 5$

Write $| 2 s - 3 | - 6 > - 5$ as a piecewise.

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

$2 s - 3 \geq 0$

Solve the inequality.

Add $3$ to both sides of the inequality.

$2 s \geq 3$

Divide each term in $2 s \geq 3$ by $2$ and simplify.

Divide each term in $2 s \geq 3$ by $2$ .

$\frac{2 s}{2} \geq \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 s}{2} \geq \frac{3}{2}$

Divide $s$ by $1$ .

$s \geq \frac{3}{2}$

In the piece where $2 s - 3$ is non-negative, remove the absolute value.

$2 s - 3 - 6 > - 5$

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

$2 s - 3 < 0$

Solve the inequality.

Add $3$ to both sides of the inequality.

$2 s < 3$

Divide each term in $2 s < 3$ by $2$ and simplify.

Divide each term in $2 s < 3$ by $2$ .

$\frac{2 s}{2} < \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 s}{2} < \frac{3}{2}$

Divide $s$ by $1$ .

$s < \frac{3}{2}$

In the piece where $2 s - 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 s - 3) - 6 > - 5$

Write as a piecewise.

${\begin{cases} 2 s - 3 - 6 > - 5 & s \geq \frac{3}{2} \\ - (2 s - 3) - 6 > - 5 & s < \frac{3}{2} \end{cases}$

Subtract $6$ from $- 3$ .

${\begin{cases} 2 s - 9 > - 5 & s \geq \frac{3}{2} \\ - (2 s - 3) - 6 > - 5 & s < \frac{3}{2} \end{cases}$

Simplify $- (2 s - 3) - 6 > - 5$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 2 s - 9 > - 5 & s \geq \frac{3}{2} \\ - (2 s) - - 3 - 6 > - 5 & s < \frac{3}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 s - 9 > - 5 & s \geq \frac{3}{2} \\ - 2 s - - 3 - 6 > - 5 & s < \frac{3}{2} \end{cases}$

Multiply $- 1$ by $- 3$ .

${\begin{cases} 2 s - 9 > - 5 & s \geq \frac{3}{2} \\ - 2 s + 3 - 6 > - 5 & s < \frac{3}{2} \end{cases}$

Subtract $6$ from $3$ .

${\begin{cases} 2 s - 9 > - 5 & s \geq \frac{3}{2} \\ - 2 s - 3 > - 5 & s < \frac{3}{2} \end{cases}$

Solve $2 s - 9 > - 5$ for $s$ .

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

Add $9$ to both sides of the inequality.

$2 s > - 5 + 9$

Add $- 5$ and $9$ .

$2 s > 4$

Divide each term in $2 s > 4$ by $2$ and simplify.

Divide each term in $2 s > 4$ by $2$ .

$\frac{2 s}{2} > \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 s}{2} > \frac{4}{2}$

Divide $s$ by $1$ .

$s > \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$s > 2$

Solve $- 2 s - 3 > - 5$ for $s$ .

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

Add $3$ to both sides of the inequality.

$- 2 s > - 5 + 3$

Add $- 5$ and $3$ .

$- 2 s > - 2$

Divide each term in $- 2 s > - 2$ by $- 2$ and simplify.

Divide each term in $- 2 s > - 2$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 s}{- 2} < \frac{- 2}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 s}{- 2} < \frac{- 2}{- 2}$

Divide $s$ by $1$ .

$s < \frac{- 2}{- 2}$

Simplify the right side.

Divide $- 2$ by $- 2$ .

$s < 1$

Find the union of the solutions.

$s < 1$ or $s > 2$

The result can be shown in multiple forms.

Inequality Form:

$s < 1 or s > 2$

Interval Notation:

$(- \infty, 1) \cup (2, \infty)$

Solve for s |2s-3|-6>-5

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