Algebra Examples

$| \frac{x}{6} | \geq 5$

Step 1

Write $| \frac{x}{6} | \geq 5$ 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.

$\frac{x}{6} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Multiply both sides by $6$ .

$\frac{x}{6} \cdot 6 \geq 0 \cdot 6$

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

Cancel the common factor of $6$ .

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Step 1.2.2.1.1.1

Cancel the common factor.

$\frac{x}{6} \cdot 6 \geq 0 \cdot 6$

Step 1.2.2.1.1.2

Rewrite the expression.

$x \geq 0 \cdot 6$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Multiply $0$ by $6$ .

$x \geq 0$

Step 1.3

In the piece where $\frac{x}{6}$ is non-negative, remove the absolute value.

$\frac{x}{6} \geq 5$

Step 1.4

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

$\frac{x}{6} < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Multiply both sides by $6$ .

$\frac{x}{6} \cdot 6 < 0 \cdot 6$

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

Cancel the common factor of $6$ .

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Step 1.5.2.1.1.1

Cancel the common factor.

$\frac{x}{6} \cdot 6 < 0 \cdot 6$

Step 1.5.2.1.1.2

Rewrite the expression.

$x < 0 \cdot 6$

Step 1.5.2.2

Simplify the right side.

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Step 1.5.2.2.1

Multiply $0$ by $6$ .

$x < 0$

Step 1.6

In the piece where $\frac{x}{6}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{x}{6} \geq 5$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{x}{6} \geq 5 & x \geq 0 \\ - \frac{x}{6} \geq 5 & x < 0 \end{cases}$

Step 2

Solve $\frac{x}{6} \geq 5$ for $x$ .

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Step 2.1

Multiply both sides by $6$ .

$\frac{x}{6} \cdot 6 \geq 5 \cdot 6$

Step 2.2

Simplify.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Cancel the common factor of $6$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{x}{6} \cdot 6 \geq 5 \cdot 6$

Step 2.2.1.1.2

Rewrite the expression.

$x \geq 5 \cdot 6$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $5$ by $6$ .

$x \geq 30$

Step 3

Solve $- \frac{x}{6} \geq 5$ for $x$ .

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Step 3.1

Divide each term in $- \frac{x}{6} \geq 5$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- \frac{x}{6} \geq 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{6}}{- 1} \leq \frac{5}{- 1}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{6}}{1} \leq \frac{5}{- 1}$

Step 3.1.2.2

Divide $\frac{x}{6}$ by $1$ .

$\frac{x}{6} \leq \frac{5}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $5$ by $- 1$ .

$\frac{x}{6} \leq - 5$

Step 3.2

Multiply both sides by $6$ .

$\frac{x}{6} \cdot 6 \leq - 5 \cdot 6$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $6$ .

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Step 3.3.1.1.1

Cancel the common factor.

$\frac{x}{6} \cdot 6 \leq - 5 \cdot 6$

Step 3.3.1.1.2

Rewrite the expression.

$x \leq - 5 \cdot 6$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $- 5$ by $6$ .

$x \leq - 30$

Step 4

Find the union of the solutions.

$x \leq - 30$ or $x \geq 30$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 30 or x \geq 30$

Interval Notation:

$(- \infty, - 30] \cup [30, \infty)$

Step 6