Solve for m 3|m-7|+9<30

$3 | m - 7 | + 9 < 30$

Step 1

Write $3 | m - 7 | + 9 < 30$ as a piecewise.

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

$m - 7 \geq 0$

Add $7$ to both sides of the inequality.

$m \geq 7$

In the piece where $m - 7$ is non-negative, remove the absolute value.

$3 (m - 7) + 9 < 30$

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

$m - 7 < 0$

Add $7$ to both sides of the inequality.

$m < 7$

In the piece where $m - 7$ is negative, remove the absolute value and multiply by $- 1$ .

$3 (- (m - 7)) + 9 < 30$

Write as a piecewise.

${\begin{cases} 3 (m - 7) + 9 < 30 & m \geq 7 \\ 3 (- (m - 7)) + 9 < 30 & m < 7 \end{cases}$

Simplify $3 (m - 7) + 9 < 30$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 3 m + 3 \cdot - 7 + 9 < 30 & m \geq 7 \\ 3 (- (m - 7)) + 9 < 30 & m < 7 \end{cases}$

Multiply $3$ by $- 7$ .

${\begin{cases} 3 m - 21 + 9 < 30 & m \geq 7 \\ 3 (- (m - 7)) + 9 < 30 & m < 7 \end{cases}$

Add $- 21$ and $9$ .

${\begin{cases} 3 m - 12 < 30 & m \geq 7 \\ 3 (- (m - 7)) + 9 < 30 & m < 7 \end{cases}$

Simplify $3 (- (m - 7)) + 9 < 30$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 3 m - 12 < 30 & m \geq 7 \\ 3 (- m - - 7) + 9 < 30 & m < 7 \end{cases}$

Multiply $- 1$ by $- 7$ .

${\begin{cases} 3 m - 12 < 30 & m \geq 7 \\ 3 (- m + 7) + 9 < 30 & m < 7 \end{cases}$

Apply the distributive property.

${\begin{cases} 3 m - 12 < 30 & m \geq 7 \\ 3 (- m) + 3 \cdot 7 + 9 < 30 & m < 7 \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} 3 m - 12 < 30 & m \geq 7 \\ - 3 m + 3 \cdot 7 + 9 < 30 & m < 7 \end{cases}$

Multiply $3$ by $7$ .

${\begin{cases} 3 m - 12 < 30 & m \geq 7 \\ - 3 m + 21 + 9 < 30 & m < 7 \end{cases}$

Add $21$ and $9$ .

${\begin{cases} 3 m - 12 < 30 & m \geq 7 \\ - 3 m + 30 < 30 & m < 7 \end{cases}$

Step 2

Solve $3 m - 12 < 30$ when $m \geq 7$ .

Solve $3 m - 12 < 30$ for $m$ .

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

Add $12$ to both sides of the inequality.

$3 m < 30 + 12$

Add $30$ and $12$ .

$3 m < 42$

Divide each term in $3 m < 42$ by $3$ and simplify.

Divide each term in $3 m < 42$ by $3$ .

$\frac{3 m}{3} < \frac{42}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 m}{3} < \frac{42}{3}$

Divide $m$ by $1$ .

$m < \frac{42}{3}$

Simplify the right side.

Divide $42$ by $3$ .

$m < 14$

Find the intersection of $m < 14$ and $m \geq 7$ .

$7 \leq m < 14$

Step 3

Solve $- 3 m + 30 < 30$ when $m < 7$ .

Solve $- 3 m + 30 < 30$ for $m$ .

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

Subtract $30$ from both sides of the inequality.

$- 3 m < 30 - 30$

Subtract $30$ from $30$ .

$- 3 m < 0$

Divide each term in $- 3 m < 0$ by $- 3$ and simplify.

Divide each term in $- 3 m < 0$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 m}{- 3} > \frac{0}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 m}{- 3} > \frac{0}{- 3}$

Divide $m$ by $1$ .

$m > \frac{0}{- 3}$

Simplify the right side.

Divide $0$ by $- 3$ .

$m > 0$

Find the intersection of $m > 0$ and $m < 7$ .

$0 < m < 7$

Step 4

Find the union of the solutions.

$0 < m < 14$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$0 < m < 14$

Interval Notation:

$(0, 14)$

Step 6

Solve for m 3|m-7|+9<30

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