Algebra Examples

$6 - 2 | m - 3 | > - 4$

Step 1

Write $6 - 2 | m - 3 | > - 4$ 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.

$m - 3 \geq 0$

Step 1.2

Add $3$ to both sides of the inequality.

$m \geq 3$

Step 1.3

In the piece where $m - 3$ is non-negative, remove the absolute value.

$6 - 2 (m - 3) > - 4$

Step 1.4

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

$m - 3 < 0$

Step 1.5

Add $3$ to both sides of the inequality.

$m < 3$

Step 1.6

In the piece where $m - 3$ is negative, remove the absolute value and multiply by $- 1$ .

$6 - 2 (- (m - 3)) > - 4$

Step 1.7

Write as a piecewise.

${\begin{cases} 6 - 2 (m - 3) > - 4 & m \geq 3 \\ 6 - 2 (- (m - 3)) > - 4 & m < 3 \end{cases}$

Step 1.8

Simplify $6 - 2 (m - 3) > - 4$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Apply the distributive property.

${\begin{cases} 6 - 2 m - 2 \cdot - 3 > - 4 & m \geq 3 \\ 6 - 2 (- (m - 3)) > - 4 & m < 3 \end{cases}$

Step 1.8.1.2

Multiply $- 2$ by $- 3$ .

${\begin{cases} 6 - 2 m + 6 > - 4 & m \geq 3 \\ 6 - 2 (- (m - 3)) > - 4 & m < 3 \end{cases}$

Step 1.8.2

Add $6$ and $6$ .

${\begin{cases} - 2 m + 12 > - 4 & m \geq 3 \\ 6 - 2 (- (m - 3)) > - 4 & m < 3 \end{cases}$

Step 1.9

Simplify $6 - 2 (- (m - 3)) > - 4$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} - 2 m + 12 > - 4 & m \geq 3 \\ 6 - 2 (- m - - 3) > - 4 & m < 3 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $- 3$ .

${\begin{cases} - 2 m + 12 > - 4 & m \geq 3 \\ 6 - 2 (- m + 3) > - 4 & m < 3 \end{cases}$

Step 1.9.1.3

Apply the distributive property.

${\begin{cases} - 2 m + 12 > - 4 & m \geq 3 \\ 6 - 2 (- m) - 2 \cdot 3 > - 4 & m < 3 \end{cases}$

Step 1.9.1.4

Multiply $- 1$ by $- 2$ .

${\begin{cases} - 2 m + 12 > - 4 & m \geq 3 \\ 6 + 2 m - 2 \cdot 3 > - 4 & m < 3 \end{cases}$

Step 1.9.1.5

Multiply $- 2$ by $3$ .

${\begin{cases} - 2 m + 12 > - 4 & m \geq 3 \\ 6 + 2 m - 6 > - 4 & m < 3 \end{cases}$

Step 1.9.2

Combine the opposite terms in $6 + 2 m - 6$ .

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Step 1.9.2.1

Subtract $6$ from $6$ .

${\begin{cases} - 2 m + 12 > - 4 & m \geq 3 \\ 2 m + 0 > - 4 & m < 3 \end{cases}$

Step 1.9.2.2

Add $2 m$ and $0$ .

${\begin{cases} - 2 m + 12 > - 4 & m \geq 3 \\ 2 m > - 4 & m < 3 \end{cases}$

Step 2

Solve $- 2 m + 12 > - 4$ when $m \geq 3$ .

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Step 2.1

Solve $- 2 m + 12 > - 4$ 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

Subtract $12$ from both sides of the inequality.

$- 2 m > - 4 - 12$

Step 2.1.1.2

Subtract $12$ from $- 4$ .

$- 2 m > - 16$

Step 2.1.2

Divide each term in $- 2 m > - 16$ by $- 2$ and simplify.

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Step 2.1.2.1

Divide each term in $- 2 m > - 16$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

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

Step 2.1.2.2.1.2

Divide $m$ by $1$ .

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

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Divide $- 16$ by $- 2$ .

$m < 8$

Step 2.2

Find the intersection of $m < 8$ and $m \geq 3$ .

$3 \leq m < 8$

Step 3

Solve $2 m > - 4$ when $m < 3$ .

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Step 3.1

Divide each term in $2 m > - 4$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 m > - 4$ by $2$ .

$\frac{2 m}{2} > \frac{- 4}{2}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $2$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{2 m}{2} > \frac{- 4}{2}$

Step 3.1.2.1.2

Divide $m$ by $1$ .

$m > \frac{- 4}{2}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $- 4$ by $2$ .

$m > - 2$

Step 3.2

Find the intersection of $m > - 2$ and $m < 3$ .

$- 2 < m < 3$

Step 4

Find the union of the solutions.

$- 2 < m < 8$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 2 < m < 8$

Interval Notation:

$(- 2, 8)$

Step 6