Solve the Absolute Value Inequality for m 3|14-m|>18

$3 | 14 - m | > 18$

Step 1

Write $3 | 14 - m | > 18$ as a piecewise.

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

$14 - m \geq 0$

Solve the inequality.

Subtract $14$ from both sides of the inequality.

$- m \geq - 14$

Divide each term in $- m \geq - 14$ by $- 1$ and simplify.

Divide each term in $- m \geq - 14$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- m}{- 1} \leq \frac{- 14}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{m}{1} \leq \frac{- 14}{- 1}$

Divide $m$ by $1$ .

$m \leq \frac{- 14}{- 1}$

Simplify the right side.

Divide $- 14$ by $- 1$ .

$m \leq 14$

In the piece where $14 - m$ is non-negative, remove the absolute value.

$3 (14 - m) > 18$

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

$14 - m < 0$

Solve the inequality.

Subtract $14$ from both sides of the inequality.

$- m < - 14$

Divide each term in $- m < - 14$ by $- 1$ and simplify.

Divide each term in $- m < - 14$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- m}{- 1} > \frac{- 14}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{m}{1} > \frac{- 14}{- 1}$

Divide $m$ by $1$ .

$m > \frac{- 14}{- 1}$

Simplify the right side.

Divide $- 14$ by $- 1$ .

$m > 14$

In the piece where $14 - m$ is negative, remove the absolute value and multiply by $- 1$ .

$3 (- (14 - m)) > 18$

Write as a piecewise.

${\begin{cases} 3 (14 - m) > 18 & m \leq 14 \\ 3 (- (14 - m)) > 18 & m > 14 \end{cases}$

Simplify $3 (14 - m) > 18$ .

Apply the distributive property.

${\begin{cases} 3 \cdot 14 + 3 (- m) > 18 & m \leq 14 \\ 3 (- (14 - m)) > 18 & m > 14 \end{cases}$

Multiply $3$ by $14$ .

${\begin{cases} 42 + 3 (- m) > 18 & m \leq 14 \\ 3 (- (14 - m)) > 18 & m > 14 \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} 42 - 3 m > 18 & m \leq 14 \\ 3 (- (14 - m)) > 18 & m > 14 \end{cases}$

Simplify $3 (- (14 - m)) > 18$ .

Apply the distributive property.

${\begin{cases} 42 - 3 m > 18 & m \leq 14 \\ 3 (- 1 \cdot 14 - - m) > 18 & m > 14 \end{cases}$

Multiply $- 1$ by $14$ .

${\begin{cases} 42 - 3 m > 18 & m \leq 14 \\ 3 (- 14 - - m) > 18 & m > 14 \end{cases}$

Multiply $- - m$ .

Multiply $- 1$ by $- 1$ .

${\begin{cases} 42 - 3 m > 18 & m \leq 14 \\ 3 (- 14 + 1 m) > 18 & m > 14 \end{cases}$

Multiply $m$ by $1$ .

${\begin{cases} 42 - 3 m > 18 & m \leq 14 \\ 3 (- 14 + m) > 18 & m > 14 \end{cases}$

Apply the distributive property.

${\begin{cases} 42 - 3 m > 18 & m \leq 14 \\ 3 \cdot - 14 + 3 m > 18 & m > 14 \end{cases}$

Multiply $3$ by $- 14$ .

${\begin{cases} 42 - 3 m > 18 & m \leq 14 \\ - 42 + 3 m > 18 & m > 14 \end{cases}$

Step 2

Solve $42 - 3 m > 18$ for $m$ .

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

Subtract $42$ from both sides of the inequality.

$- 3 m > 18 - 42$

Subtract $42$ from $18$ .

$- 3 m > - 24$

Divide each term in $- 3 m > - 24$ by $- 3$ and simplify.

Divide each term in $- 3 m > - 24$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 m}{- 3} < \frac{- 24}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 m}{- 3} < \frac{- 24}{- 3}$

Divide $m$ by $1$ .

$m < \frac{- 24}{- 3}$

Simplify the right side.

Divide $- 24$ by $- 3$ .

$m < 8$

Step 3

Solve $- 42 + 3 m > 18$ for $m$ .

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

Add $42$ to both sides of the inequality.

$3 m > 18 + 42$

Add $18$ and $42$ .

$3 m > 60$

Divide each term in $3 m > 60$ by $3$ and simplify.

Divide each term in $3 m > 60$ by $3$ .

$\frac{3 m}{3} > \frac{60}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 m}{3} > \frac{60}{3}$

Divide $m$ by $1$ .

$m > \frac{60}{3}$

Simplify the right side.

Divide $60$ by $3$ .

$m > 20$

Step 4

Find the union of the solutions.

$m < 8$ or $m > 20$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$m < 8 or m > 20$

Interval Notation:

$(- \infty, 8) \cup (20, \infty)$

Step 6

Solve the Absolute Value Inequality for m 3|14-m|>18

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