Algebra Examples

$| 2 a | - 4 \leq 8$

Step 1

Solve $- 6 \leq a \leq 6$ .

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Step 1.1

Write $| 2 a | - 4 \leq 8$ as a piecewise.

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Step 1.1.1

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

$2 a \geq 0$

Step 1.1.2

Divide each term in $2 a \geq 0$ by $2$ and simplify.

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Step 1.1.2.1

Divide each term in $2 a \geq 0$ by $2$ .

$\frac{2 a}{2} \geq \frac{0}{2}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{2 a}{2} \geq \frac{0}{2}$

Step 1.1.2.2.1.2

Divide $a$ by $1$ .

$a \geq \frac{0}{2}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Divide $0$ by $2$ .

$a \geq 0$

Step 1.1.3

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

$2 a - 4 \leq 8$

Step 1.1.4

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

$2 a < 0$

Step 1.1.5

Divide each term in $2 a < 0$ by $2$ and simplify.

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Step 1.1.5.1

Divide each term in $2 a < 0$ by $2$ .

$\frac{2 a}{2} < \frac{0}{2}$

Step 1.1.5.2

Simplify the left side.

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Step 1.1.5.2.1

Cancel the common factor of $2$ .

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Step 1.1.5.2.1.1

Cancel the common factor.

$\frac{2 a}{2} < \frac{0}{2}$

Step 1.1.5.2.1.2

Divide $a$ by $1$ .

$a < \frac{0}{2}$

Step 1.1.5.3

Simplify the right side.

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Step 1.1.5.3.1

Divide $0$ by $2$ .

$a < 0$

Step 1.1.6

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

$- 2 a - 4 \leq 8$

Step 1.1.7

Write as a piecewise.

${\begin{cases} 2 a - 4 \leq 8 & a \geq 0 \\ - 2 a - 4 \leq 8 & a < 0 \end{cases}$

Step 1.2

Solve $2 a - 4 \leq 8$ when $a \geq 0$ .

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Step 1.2.1

Solve $2 a - 4 \leq 8$ for $a$ .

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Step 1.2.1.1

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

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Step 1.2.1.1.1

Add $4$ to both sides of the inequality.

$2 a \leq 8 + 4$

Step 1.2.1.1.2

Add $8$ and $4$ .

$2 a \leq 12$

Step 1.2.1.2

Divide each term in $2 a \leq 12$ by $2$ and simplify.

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Step 1.2.1.2.1

Divide each term in $2 a \leq 12$ by $2$ .

$\frac{2 a}{2} \leq \frac{12}{2}$

Step 1.2.1.2.2

Simplify the left side.

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Step 1.2.1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.2.2.1.1

Cancel the common factor.

$\frac{2 a}{2} \leq \frac{12}{2}$

Step 1.2.1.2.2.1.2

Divide $a$ by $1$ .

$a \leq \frac{12}{2}$

Step 1.2.1.2.3

Simplify the right side.

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Step 1.2.1.2.3.1

Divide $12$ by $2$ .

$a \leq 6$

Step 1.2.2

Find the intersection of $a \leq 6$ and $a \geq 0$ .

$0 \leq a \leq 6$

Step 1.3

Solve $- 2 a - 4 \leq 8$ when $a < 0$ .

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Step 1.3.1

Solve $- 2 a - 4 \leq 8$ for $a$ .

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Step 1.3.1.1

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

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Step 1.3.1.1.1

Add $4$ to both sides of the inequality.

$- 2 a \leq 8 + 4$

Step 1.3.1.1.2

Add $8$ and $4$ .

$- 2 a \leq 12$

Step 1.3.1.2

Divide each term in $- 2 a \leq 12$ by $- 2$ and simplify.

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Step 1.3.1.2.1

Divide each term in $- 2 a \leq 12$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 a}{- 2} \geq \frac{12}{- 2}$

Step 1.3.1.2.2

Simplify the left side.

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Step 1.3.1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.3.1.2.2.1.1

Cancel the common factor.

$\frac{- 2 a}{- 2} \geq \frac{12}{- 2}$

Step 1.3.1.2.2.1.2

Divide $a$ by $1$ .

$a \geq \frac{12}{- 2}$

Step 1.3.1.2.3

Simplify the right side.

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Step 1.3.1.2.3.1

Divide $12$ by $- 2$ .

$a \geq - 6$

Step 1.3.2

Find the intersection of $a \geq - 6$ and $a < 0$ .

$- 6 \leq a < 0$

Step 1.4

Find the union of the solutions.

$- 6 \leq a \leq 6$

Step 2

Use the inequality $- 6 \leq a \leq 6$ to build the set notation.

${a | - 6 \leq a \leq 6}$

Step 3