Algebra Examples

$| 8 x | \geq 16$

Step 1

Write $| 8 x | \geq 16$ 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.

$8 x \geq 0$

Step 1.2

Divide each term in $8 x \geq 0$ by $8$ and simplify.

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Step 1.2.1

Divide each term in $8 x \geq 0$ by $8$ .

$\frac{8 x}{8} \geq \frac{0}{8}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $8$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} \geq \frac{0}{8}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{8}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $8$ .

$x \geq 0$

Step 1.3

In the piece where $8 x$ is non-negative, remove the absolute value.

$8 x \geq 16$

Step 1.4

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

$8 x < 0$

Step 1.5

Divide each term in $8 x < 0$ by $8$ and simplify.

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Step 1.5.1

Divide each term in $8 x < 0$ by $8$ .

$\frac{8 x}{8} < \frac{0}{8}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $8$ .

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Step 1.5.2.1.1

Cancel the common factor.

$\frac{8 x}{8} < \frac{0}{8}$

Step 1.5.2.1.2

Divide $x$ by $1$ .

$x < \frac{0}{8}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Divide $0$ by $8$ .

$x < 0$

Step 1.6

In the piece where $8 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- 8 x \geq 16$

Step 1.7

Write as a piecewise.

${\begin{cases} 8 x \geq 16 & x \geq 0 \\ - 8 x \geq 16 & x < 0 \end{cases}$

Step 2

Divide each term in $8 x \geq 16$ by $8$ and simplify.

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Step 2.1

Divide each term in $8 x \geq 16$ by $8$ .

$\frac{8 x}{8} \geq \frac{16}{8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $8$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} \geq \frac{16}{8}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{16}{8}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $16$ by $8$ .

$x \geq 2$

Step 3

Divide each term in $- 8 x \geq 16$ by $- 8$ and simplify.

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Step 3.1

Divide each term in $- 8 x \geq 16$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} \leq \frac{16}{- 8}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 8$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} \leq \frac{16}{- 8}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{16}{- 8}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $16$ by $- 8$ .

$x \leq - 2$

Step 4

Find the union of the solutions.

$x \leq - 2$ or $x \geq 2$

Step 5