Algebra Examples

$2 | x + 4 | - 6 \leq 8$

Step 1

Write $2 | x + 4 | - 6 \leq 8$ 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.

$x + 4 \geq 0$

Step 1.2

Subtract $4$ from both sides of the inequality.

$x \geq - 4$

Step 1.3

In the piece where $x + 4$ is non-negative, remove the absolute value.

$2 (x + 4) - 6 \leq 8$

Step 1.4

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

$x + 4 < 0$

Step 1.5

Subtract $4$ from both sides of the inequality.

$x < - 4$

Step 1.6

In the piece where $x + 4$ is negative, remove the absolute value and multiply by $- 1$ .

$2 (- (x + 4)) - 6 \leq 8$

Step 1.7

Write as a piecewise.

${\begin{cases} 2 (x + 4) - 6 \leq 8 & x \geq - 4 \\ 2 (- (x + 4)) - 6 \leq 8 & x < - 4 \end{cases}$

Step 1.8

Simplify $2 (x + 4) - 6 \leq 8$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Apply the distributive property.

${\begin{cases} 2 x + 2 \cdot 4 - 6 \leq 8 & x \geq - 4 \\ 2 (- (x + 4)) - 6 \leq 8 & x < - 4 \end{cases}$

Step 1.8.1.2

Multiply $2$ by $4$ .

${\begin{cases} 2 x + 8 - 6 \leq 8 & x \geq - 4 \\ 2 (- (x + 4)) - 6 \leq 8 & x < - 4 \end{cases}$

Step 1.8.2

Subtract $6$ from $8$ .

${\begin{cases} 2 x + 2 \leq 8 & x \geq - 4 \\ 2 (- (x + 4)) - 6 \leq 8 & x < - 4 \end{cases}$

Step 1.9

Simplify $2 (- (x + 4)) - 6 \leq 8$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 2 x + 2 \leq 8 & x \geq - 4 \\ 2 (- x - 1 \cdot 4) - 6 \leq 8 & x < - 4 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $4$ .

${\begin{cases} 2 x + 2 \leq 8 & x \geq - 4 \\ 2 (- x - 4) - 6 \leq 8 & x < - 4 \end{cases}$

Step 1.9.1.3

Apply the distributive property.

${\begin{cases} 2 x + 2 \leq 8 & x \geq - 4 \\ 2 (- x) + 2 \cdot - 4 - 6 \leq 8 & x < - 4 \end{cases}$

Step 1.9.1.4

Multiply $- 1$ by $2$ .

${\begin{cases} 2 x + 2 \leq 8 & x \geq - 4 \\ - 2 x + 2 \cdot - 4 - 6 \leq 8 & x < - 4 \end{cases}$

Step 1.9.1.5

Multiply $2$ by $- 4$ .

${\begin{cases} 2 x + 2 \leq 8 & x \geq - 4 \\ - 2 x - 8 - 6 \leq 8 & x < - 4 \end{cases}$

Step 1.9.2

Subtract $6$ from $- 8$ .

${\begin{cases} 2 x + 2 \leq 8 & x \geq - 4 \\ - 2 x - 14 \leq 8 & x < - 4 \end{cases}$

Step 2

Solve $2 x + 2 \leq 8$ when $x \geq - 4$ .

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Step 2.1

Solve $2 x + 2 \leq 8$ for $x$ .

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Step 2.1.1

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

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Step 2.1.1.1

Subtract $2$ from both sides of the inequality.

$2 x \leq 8 - 2$

Step 2.1.1.2

Subtract $2$ from $8$ .

$2 x \leq 6$

Step 2.1.2

Divide each term in $2 x \leq 6$ by $2$ and simplify.

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Step 2.1.2.1

Divide each term in $2 x \leq 6$ by $2$ .

$\frac{2 x}{2} \leq \frac{6}{2}$

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.

$\frac{2 x}{2} \leq \frac{6}{2}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{6}{2}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Divide $6$ by $2$ .

$x \leq 3$

Step 2.2

Find the intersection of $x \leq 3$ and $x \geq - 4$ .

$- 4 \leq x \leq 3$

Step 3

Solve $- 2 x - 14 \leq 8$ when $x < - 4$ .

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Step 3.1

Solve $- 2 x - 14 \leq 8$ for $x$ .

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Step 3.1.1

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

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Step 3.1.1.1

Add $14$ to both sides of the inequality.

$- 2 x \leq 8 + 14$

Step 3.1.1.2

Add $8$ and $14$ .

$- 2 x \leq 22$

Step 3.1.2

Divide each term in $- 2 x \leq 22$ by $- 2$ and simplify.

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Step 3.1.2.1

Divide each term in $- 2 x \leq 22$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \geq \frac{22}{- 2}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \geq \frac{22}{- 2}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{22}{- 2}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $22$ by $- 2$ .

$x \geq - 11$

Step 3.2

Find the intersection of $x \geq - 11$ and $x < - 4$ .

$- 11 \leq x < - 4$

Step 4

Find the union of the solutions.

$- 11 \leq x \leq 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 11 \leq x \leq 3$

Interval Notation:

$[- 11, 3]$

Step 6