Solve the Inequality for m -15+6|2m-1|<3

$- 15 + 6 | 2 m - 1 | < 3$

Step 1

Write $- 15 + 6 | 2 m - 1 | < 3$ as a piecewise.

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

$2 m - 1 \geq 0$

Solve the inequality.

Add $1$ to both sides of the inequality.

$2 m \geq 1$

Divide each term in $2 m \geq 1$ by $2$ and simplify.

Divide each term in $2 m \geq 1$ by $2$ .

$\frac{2 m}{2} \geq \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 m}{2} \geq \frac{1}{2}$

Divide $m$ by $1$ .

$m \geq \frac{1}{2}$

In the piece where $2 m - 1$ is non-negative, remove the absolute value.

$- 15 + 6 (2 m - 1) < 3$

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

$2 m - 1 < 0$

Solve the inequality.

Add $1$ to both sides of the inequality.

$2 m < 1$

Divide each term in $2 m < 1$ by $2$ and simplify.

Divide each term in $2 m < 1$ by $2$ .

$\frac{2 m}{2} < \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 m}{2} < \frac{1}{2}$

Divide $m$ by $1$ .

$m < \frac{1}{2}$

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

$- 15 + 6 (- (2 m - 1)) < 3$

Write as a piecewise.

${\begin{cases} - 15 + 6 (2 m - 1) < 3 & m \geq \frac{1}{2} \\ - 15 + 6 (- (2 m - 1)) < 3 & m < \frac{1}{2} \end{cases}$

Simplify $- 15 + 6 (2 m - 1) < 3$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 15 + 6 (2 m) + 6 \cdot - 1 < 3 & m \geq \frac{1}{2} \\ - 15 + 6 (- (2 m - 1)) < 3 & m < \frac{1}{2} \end{cases}$

Multiply $2$ by $6$ .

${\begin{cases} - 15 + 12 m + 6 \cdot - 1 < 3 & m \geq \frac{1}{2} \\ - 15 + 6 (- (2 m - 1)) < 3 & m < \frac{1}{2} \end{cases}$

Multiply $6$ by $- 1$ .

${\begin{cases} - 15 + 12 m - 6 < 3 & m \geq \frac{1}{2} \\ - 15 + 6 (- (2 m - 1)) < 3 & m < \frac{1}{2} \end{cases}$

Subtract $6$ from $- 15$ .

${\begin{cases} 12 m - 21 < 3 & m \geq \frac{1}{2} \\ - 15 + 6 (- (2 m - 1)) < 3 & m < \frac{1}{2} \end{cases}$

Simplify $- 15 + 6 (- (2 m - 1)) < 3$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 12 m - 21 < 3 & m \geq \frac{1}{2} \\ - 15 + 6 (- (2 m) - - 1) < 3 & m < \frac{1}{2} \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 12 m - 21 < 3 & m \geq \frac{1}{2} \\ - 15 + 6 (- 2 m - - 1) < 3 & m < \frac{1}{2} \end{cases}$

Multiply $- 1$ by $- 1$ .

${\begin{cases} 12 m - 21 < 3 & m \geq \frac{1}{2} \\ - 15 + 6 (- 2 m + 1) < 3 & m < \frac{1}{2} \end{cases}$

Apply the distributive property.

${\begin{cases} 12 m - 21 < 3 & m \geq \frac{1}{2} \\ - 15 + 6 (- 2 m) + 6 \cdot 1 < 3 & m < \frac{1}{2} \end{cases}$

Multiply $- 2$ by $6$ .

${\begin{cases} 12 m - 21 < 3 & m \geq \frac{1}{2} \\ - 15 - 12 m + 6 \cdot 1 < 3 & m < \frac{1}{2} \end{cases}$

Multiply $6$ by $1$ .

${\begin{cases} 12 m - 21 < 3 & m \geq \frac{1}{2} \\ - 15 - 12 m + 6 < 3 & m < \frac{1}{2} \end{cases}$

Add $- 15$ and $6$ .

${\begin{cases} 12 m - 21 < 3 & m \geq \frac{1}{2} \\ - 12 m - 9 < 3 & m < \frac{1}{2} \end{cases}$

Step 2

Solve $12 m - 21 < 3$ when $m \geq \frac{1}{2}$ .

Solve $12 m - 21 < 3$ for $m$ .

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

Add $21$ to both sides of the inequality.

$12 m < 3 + 21$

Add $3$ and $21$ .

$12 m < 24$

Divide each term in $12 m < 24$ by $12$ and simplify.

Divide each term in $12 m < 24$ by $12$ .

$\frac{12 m}{12} < \frac{24}{12}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 m}{12} < \frac{24}{12}$

Divide $m$ by $1$ .

$m < \frac{24}{12}$

Simplify the right side.

Divide $24$ by $12$ .

$m < 2$

Find the intersection of $m < 2$ and $m \geq \frac{1}{2}$ .

$\frac{1}{2} \leq m < 2$

Step 3

Solve $- 12 m - 9 < 3$ when $m < \frac{1}{2}$ .

Solve $- 12 m - 9 < 3$ for $m$ .

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

Add $9$ to both sides of the inequality.

$- 12 m < 3 + 9$

Add $3$ and $9$ .

$- 12 m < 12$

Divide each term in $- 12 m < 12$ by $- 12$ and simplify.

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

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

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

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

Divide $m$ by $1$ .

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

Simplify the right side.

Divide $12$ by $- 12$ .

$m > - 1$

Find the intersection of $m > - 1$ and $m < \frac{1}{2}$ .

$- 1 < m < \frac{1}{2}$

Step 4

Find the union of the solutions.

$- 1 < m < 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 1 < m < 2$

Interval Notation:

$(- 1, 2)$

Step 6

Solve the Inequality for m -15+6|2m-1|<3

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