Algebra Examples

$| \frac{m + 3 m}{3} | - 1 \leq 2$

Step 1

Write $| \frac{m + 3 m}{3} | - 1 \leq 2$ as a piecewise.

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Step 1.1

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

$\frac{m + 3 m}{3} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Multiply both sides by $3$ .

$\frac{m + 3 m}{3} \cdot 3 \geq 0 \cdot 3$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Simplify the left side.

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Step 1.2.2.1.1

Simplify $\frac{m + 3 m}{3} \cdot 3$ .

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Step 1.2.2.1.1.1

Add $m$ and $3 m$ .

$\frac{4 m}{3} \cdot 3 \geq 0 \cdot 3$

Step 1.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.2.1.1.2.1

Cancel the common factor.

$\frac{4 m}{3} \cdot 3 \geq 0 \cdot 3$

Step 1.2.2.1.1.2.2

Rewrite the expression.

$4 m \geq 0 \cdot 3$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Multiply $0$ by $3$ .

$4 m \geq 0$

Step 1.2.3

Divide each term in $4 m \geq 0$ by $4$ and simplify.

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Step 1.2.3.1

Divide each term in $4 m \geq 0$ by $4$ .

$\frac{4 m}{4} \geq \frac{0}{4}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $4$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{4 m}{4} \geq \frac{0}{4}$

Step 1.2.3.2.1.2

Divide $m$ by $1$ .

$m \geq \frac{0}{4}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $4$ .

$m \geq 0$

Step 1.3

In the piece where $\frac{m + 3 m}{3}$ is non-negative, remove the absolute value.

$\frac{m + 3 m}{3} - 1 \leq 2$

Step 1.4

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

$\frac{m + 3 m}{3} < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Multiply both sides by $3$ .

$\frac{m + 3 m}{3} \cdot 3 < 0 \cdot 3$

Step 1.5.2

Simplify.

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Step 1.5.2.1

Simplify the left side.

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Step 1.5.2.1.1

Simplify $\frac{m + 3 m}{3} \cdot 3$ .

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Step 1.5.2.1.1.1

Add $m$ and $3 m$ .

$\frac{4 m}{3} \cdot 3 < 0 \cdot 3$

Step 1.5.2.1.1.2

Cancel the common factor of $3$ .

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Step 1.5.2.1.1.2.1

Cancel the common factor.

$\frac{4 m}{3} \cdot 3 < 0 \cdot 3$

Step 1.5.2.1.1.2.2

Rewrite the expression.

$4 m < 0 \cdot 3$

Step 1.5.2.2

Simplify the right side.

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Step 1.5.2.2.1

Multiply $0$ by $3$ .

$4 m < 0$

Step 1.5.3

Divide each term in $4 m < 0$ by $4$ and simplify.

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Step 1.5.3.1

Divide each term in $4 m < 0$ by $4$ .

$\frac{4 m}{4} < \frac{0}{4}$

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Cancel the common factor of $4$ .

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Step 1.5.3.2.1.1

Cancel the common factor.

$\frac{4 m}{4} < \frac{0}{4}$

Step 1.5.3.2.1.2

Divide $m$ by $1$ .

$m < \frac{0}{4}$

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

Divide $0$ by $4$ .

$m < 0$

Step 1.6

In the piece where $\frac{m + 3 m}{3}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{m + 3 m}{3} - 1 \leq 2$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{m + 3 m}{3} - 1 \leq 2 & m \geq 0 \\ - \frac{m + 3 m}{3} - 1 \leq 2 & m < 0 \end{cases}$

Step 1.8

Add $m$ and $3 m$ .

${\begin{cases} \frac{4 m}{3} - 1 \leq 2 & m \geq 0 \\ - \frac{m + 3 m}{3} - 1 \leq 2 & m < 0 \end{cases}$

Step 1.9

Add $m$ and $3 m$ .

${\begin{cases} \frac{4 m}{3} - 1 \leq 2 & m \geq 0 \\ - \frac{4 m}{3} - 1 \leq 2 & m < 0 \end{cases}$

Step 2

Solve $\frac{4 m}{3} - 1 \leq 2$ when $m \geq 0$ .

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Step 2.1

Solve $\frac{4 m}{3} - 1 \leq 2$ for $m$ .

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Step 2.1.1

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

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Step 2.1.1.1

Add $1$ to both sides of the inequality.

$\frac{4 m}{3} \leq 2 + 1$

Step 2.1.1.2

Add $2$ and $1$ .

$\frac{4 m}{3} \leq 3$

Step 2.1.2

Multiply both sides by $3$ .

$\frac{4 m}{3} \cdot 3 \leq 3 \cdot 3$

Step 2.1.3

Simplify.

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Step 2.1.3.1

Simplify the left side.

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Step 2.1.3.1.1

Cancel the common factor of $3$ .

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Step 2.1.3.1.1.1

Cancel the common factor.

$\frac{4 m}{3} \cdot 3 \leq 3 \cdot 3$

Step 2.1.3.1.1.2

Rewrite the expression.

$4 m \leq 3 \cdot 3$

Step 2.1.3.2

Simplify the right side.

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Step 2.1.3.2.1

Multiply $3$ by $3$ .

$4 m \leq 9$

Step 2.1.4

Divide each term in $4 m \leq 9$ by $4$ and simplify.

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Step 2.1.4.1

Divide each term in $4 m \leq 9$ by $4$ .

$\frac{4 m}{4} \leq \frac{9}{4}$

Step 2.1.4.2

Simplify the left side.

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Step 2.1.4.2.1

Cancel the common factor of $4$ .

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Step 2.1.4.2.1.1

Cancel the common factor.

$\frac{4 m}{4} \leq \frac{9}{4}$

Step 2.1.4.2.1.2

Divide $m$ by $1$ .

$m \leq \frac{9}{4}$

Step 2.2

Find the intersection of $m \leq \frac{9}{4}$ and $m \geq 0$ .

$0 \leq m \leq \frac{9}{4}$

Step 3

Solve $- \frac{4 m}{3} - 1 \leq 2$ when $m < 0$ .

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Step 3.1

Solve $- \frac{4 m}{3} - 1 \leq 2$ for $m$ .

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Step 3.1.1

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

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Step 3.1.1.1

Add $1$ to both sides of the inequality.

$- \frac{4 m}{3} \leq 2 + 1$

Step 3.1.1.2

Add $2$ and $1$ .

$- \frac{4 m}{3} \leq 3$

Step 3.1.2

Divide each term in $- \frac{4 m}{3} \leq 3$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- \frac{4 m}{3} \leq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{4 m}{3}}{- 1} \geq \frac{3}{- 1}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{4 m}{3}}{1} \geq \frac{3}{- 1}$

Step 3.1.2.2.2

Divide $\frac{4 m}{3}$ by $1$ .

$\frac{4 m}{3} \geq \frac{3}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $3$ by $- 1$ .

$\frac{4 m}{3} \geq - 3$

Step 3.1.3

Multiply both sides by $3$ .

$\frac{4 m}{3} \cdot 3 \geq - 3 \cdot 3$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Simplify the left side.

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Step 3.1.4.1.1

Cancel the common factor of $3$ .

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Step 3.1.4.1.1.1

Cancel the common factor.

$\frac{4 m}{3} \cdot 3 \geq - 3 \cdot 3$

Step 3.1.4.1.1.2

Rewrite the expression.

$4 m \geq - 3 \cdot 3$

Step 3.1.4.2

Simplify the right side.

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Step 3.1.4.2.1

Multiply $- 3$ by $3$ .

$4 m \geq - 9$

Step 3.1.5

Divide each term in $4 m \geq - 9$ by $4$ and simplify.

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Step 3.1.5.1

Divide each term in $4 m \geq - 9$ by $4$ .

$\frac{4 m}{4} \geq \frac{- 9}{4}$

Step 3.1.5.2

Simplify the left side.

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Step 3.1.5.2.1

Cancel the common factor of $4$ .

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Step 3.1.5.2.1.1

Cancel the common factor.

$\frac{4 m}{4} \geq \frac{- 9}{4}$

Step 3.1.5.2.1.2

Divide $m$ by $1$ .

$m \geq \frac{- 9}{4}$

Step 3.1.5.3

Simplify the right side.

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Step 3.1.5.3.1

Move the negative in front of the fraction.

$m \geq - \frac{9}{4}$

Step 3.2

Find the intersection of $m \geq - \frac{9}{4}$ and $m < 0$ .

$- \frac{9}{4} \leq m < 0$

Step 4

Find the union of the solutions.

$- \frac{9}{4} \leq m \leq \frac{9}{4}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- \frac{9}{4} \leq m \leq \frac{9}{4}$

Interval Notation:

$[- \frac{9}{4}, \frac{9}{4}]$

Step 6