Solve the Inequality for m |2-(5m)/2|<14

$| 2 - \frac{5 m}{2} | < 14$

Write $| 2 - \frac{5 m}{2} | < 14$ as a piecewise.

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

$2 - \frac{5 m}{2} \geq 0$

Solve the inequality.

Subtract $2$ from both sides of the inequality.

$- \frac{5 m}{2} \geq - 2$

Divide each term in $- \frac{5 m}{2} \geq - 2$ by $- 1$ and simplify.

Divide each term in $- \frac{5 m}{2} \geq - 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{5 m}{2}}{- 1} \leq \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{5 m}{2}}{1} \leq \frac{- 2}{- 1}$

Divide $\frac{5 m}{2}$ by $1$ .

$\frac{5 m}{2} \leq \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$\frac{5 m}{2} \leq 2$

Multiply both sides by $2$ .

$\frac{5 m}{2} \cdot 2 \leq 2 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 m}{2} \cdot 2 \leq 2 \cdot 2$

Rewrite the expression.

$5 m \leq 2 \cdot 2$

Simplify the right side.

Multiply $2$ by $2$ .

$5 m \leq 4$

Divide each term in $5 m \leq 4$ by $5$ and simplify.

Divide each term in $5 m \leq 4$ by $5$ .

$\frac{5 m}{5} \leq \frac{4}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 m}{5} \leq \frac{4}{5}$

Divide $m$ by $1$ .

$m \leq \frac{4}{5}$

In the piece where $2 - \frac{5 m}{2}$ is non-negative, remove the absolute value.

$2 - \frac{5 m}{2} < 14$

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

$2 - \frac{5 m}{2} < 0$

Solve the inequality.

Subtract $2$ from both sides of the inequality.

$- \frac{5 m}{2} < - 2$

Divide each term in $- \frac{5 m}{2} < - 2$ by $- 1$ and simplify.

Divide each term in $- \frac{5 m}{2} < - 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{5 m}{2}}{- 1} > \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{5 m}{2}}{1} > \frac{- 2}{- 1}$

Divide $\frac{5 m}{2}$ by $1$ .

$\frac{5 m}{2} > \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$\frac{5 m}{2} > 2$

Multiply both sides by $2$ .

$\frac{5 m}{2} \cdot 2 > 2 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 m}{2} \cdot 2 > 2 \cdot 2$

Rewrite the expression.

$5 m > 2 \cdot 2$

Simplify the right side.

Multiply $2$ by $2$ .

$5 m > 4$

Divide each term in $5 m > 4$ by $5$ and simplify.

Divide each term in $5 m > 4$ by $5$ .

$\frac{5 m}{5} > \frac{4}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 m}{5} > \frac{4}{5}$

Divide $m$ by $1$ .

$m > \frac{4}{5}$

In the piece where $2 - \frac{5 m}{2}$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 - \frac{5 m}{2}) < 14$

Write as a piecewise.

${\begin{cases} 2 - \frac{5 m}{2} < 14 & m \leq \frac{4}{5} \\ - (2 - \frac{5 m}{2}) < 14 & m > \frac{4}{5} \end{cases}$

Simplify $- (2 - \frac{5 m}{2}) < 14$ .

Apply the distributive property.

${\begin{cases} 2 - \frac{5 m}{2} < 14 & m \leq \frac{4}{5} \\ - 1 \cdot 2 - - \frac{5 m}{2} < 14 & m > \frac{4}{5} \end{cases}$

Multiply $- 1$ by $2$ .

${\begin{cases} 2 - \frac{5 m}{2} < 14 & m \leq \frac{4}{5} \\ - 2 - - \frac{5 m}{2} < 14 & m > \frac{4}{5} \end{cases}$

Multiply $- - \frac{5 m}{2}$ .

Multiply $- 1$ by $- 1$ .

${\begin{cases} 2 - \frac{5 m}{2} < 14 & m \leq \frac{4}{5} \\ - 2 + 1 \frac{5 m}{2} < 14 & m > \frac{4}{5} \end{cases}$

Multiply $\frac{5 m}{2}$ by $1$ .

${\begin{cases} 2 - \frac{5 m}{2} < 14 & m \leq \frac{4}{5} \\ - 2 + \frac{5 m}{2} < 14 & m > \frac{4}{5} \end{cases}$

Solve $2 - \frac{5 m}{2} < 14$ when $m \leq \frac{4}{5}$ .

Solve $2 - \frac{5 m}{2} < 14$ for $m$ .

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

Subtract $2$ from both sides of the inequality.

$- \frac{5 m}{2} < 14 - 2$

Subtract $2$ from $14$ .

$- \frac{5 m}{2} < 12$

Divide each term in $- \frac{5 m}{2} < 12$ by $- 1$ and simplify.

Divide each term in $- \frac{5 m}{2} < 12$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{5 m}{2}}{- 1} > \frac{12}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{5 m}{2}}{1} > \frac{12}{- 1}$

Divide $\frac{5 m}{2}$ by $1$ .

$\frac{5 m}{2} > \frac{12}{- 1}$

Simplify the right side.

Divide $12$ by $- 1$ .

$\frac{5 m}{2} > - 12$

Multiply both sides by $2$ .

$\frac{5 m}{2} \cdot 2 > - 12 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 m}{2} \cdot 2 > - 12 \cdot 2$

Rewrite the expression.

$5 m > - 12 \cdot 2$

Simplify the right side.

Multiply $- 12$ by $2$ .

$5 m > - 24$

Divide each term in $5 m > - 24$ by $5$ and simplify.

Divide each term in $5 m > - 24$ by $5$ .

$\frac{5 m}{5} > \frac{- 24}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 m}{5} > \frac{- 24}{5}$

Divide $m$ by $1$ .

$m > \frac{- 24}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$m > - \frac{24}{5}$

Find the intersection of $m > - \frac{24}{5}$ and $m \leq \frac{4}{5}$ .

$- \frac{24}{5} < m \leq \frac{4}{5}$

Solve $- 2 + \frac{5 m}{2} < 14$ when $m > \frac{4}{5}$ .

Solve $- 2 + \frac{5 m}{2} < 14$ for $m$ .

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

Add $2$ to both sides of the inequality.

$\frac{5 m}{2} < 14 + 2$

Add $14$ and $2$ .

$\frac{5 m}{2} < 16$

Multiply both sides by $2$ .

$\frac{5 m}{2} \cdot 2 < 16 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 m}{2} \cdot 2 < 16 \cdot 2$

Rewrite the expression.

$5 m < 16 \cdot 2$

Simplify the right side.

Multiply $16$ by $2$ .

$5 m < 32$

Divide each term in $5 m < 32$ by $5$ and simplify.

Divide each term in $5 m < 32$ by $5$ .

$\frac{5 m}{5} < \frac{32}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 m}{5} < \frac{32}{5}$

Divide $m$ by $1$ .

$m < \frac{32}{5}$

Find the intersection of $m < \frac{32}{5}$ and $m > \frac{4}{5}$ .

$\frac{4}{5} < m < \frac{32}{5}$

Find the union of the solutions.

$- \frac{24}{5} < m < \frac{32}{5}$

The result can be shown in multiple forms.

Inequality Form:

$- \frac{24}{5} < m < \frac{32}{5}$

Interval Notation:

$(- \frac{24}{5}, \frac{32}{5})$

Solve the Inequality for m |2-(5m)/2|<14

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