Precalculus Examples

$| 2 - 5 x | - 8 > 17$

Step 1

Write $| 2 - 5 x | - 8 > 17$ 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.

$2 - 5 x \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Subtract $2$ from both sides of the inequality.

$- 5 x \geq - 2$

Step 1.2.2

Divide each term in $- 5 x \geq - 2$ by $- 5$ and simplify.

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Step 1.2.2.1

Divide each term in $- 5 x \geq - 2$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} \leq \frac{- 2}{- 5}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $- 5$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} \leq \frac{- 2}{- 5}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 2}{- 5}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Dividing two negative values results in a positive value.

$x \leq \frac{2}{5}$

Step 1.3

In the piece where $2 - 5 x$ is non-negative, remove the absolute value.

$2 - 5 x - 8 > 17$

Step 1.4

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

$2 - 5 x < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Subtract $2$ from both sides of the inequality.

$- 5 x < - 2$

Step 1.5.2

Divide each term in $- 5 x < - 2$ by $- 5$ and simplify.

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Step 1.5.2.1

Divide each term in $- 5 x < - 2$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} > \frac{- 2}{- 5}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $- 5$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} > \frac{- 2}{- 5}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 2}{- 5}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Dividing two negative values results in a positive value.

$x > \frac{2}{5}$

Step 1.6

In the piece where $2 - 5 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 - 5 x) - 8 > 17$

Step 1.7

Write as a piecewise.

${\begin{cases} 2 - 5 x - 8 > 17 & x \leq \frac{2}{5} \\ - (2 - 5 x) - 8 > 17 & x > \frac{2}{5} \end{cases}$

Step 1.8

Subtract $8$ from $2$ .

${\begin{cases} - 5 x - 6 > 17 & x \leq \frac{2}{5} \\ - (2 - 5 x) - 8 > 17 & x > \frac{2}{5} \end{cases}$

Step 1.9

Simplify $- (2 - 5 x) - 8 > 17$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} - 5 x - 6 > 17 & x \leq \frac{2}{5} \\ - 1 \cdot 2 - (- 5 x) - 8 > 17 & x > \frac{2}{5} \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $2$ .

${\begin{cases} - 5 x - 6 > 17 & x \leq \frac{2}{5} \\ - 2 - (- 5 x) - 8 > 17 & x > \frac{2}{5} \end{cases}$

Step 1.9.1.3

Multiply $- 5$ by $- 1$ .

${\begin{cases} - 5 x - 6 > 17 & x \leq \frac{2}{5} \\ - 2 + 5 x - 8 > 17 & x > \frac{2}{5} \end{cases}$

Step 1.9.2

Subtract $8$ from $- 2$ .

${\begin{cases} - 5 x - 6 > 17 & x \leq \frac{2}{5} \\ 5 x - 10 > 17 & x > \frac{2}{5} \end{cases}$

Step 2

Solve $- 5 x - 6 > 17$ for $x$ .

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Step 2.1

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

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Step 2.1.1

Add $6$ to both sides of the inequality.

$- 5 x > 17 + 6$

Step 2.1.2

Add $17$ and $6$ .

$- 5 x > 23$

Step 2.2

Divide each term in $- 5 x > 23$ by $- 5$ and simplify.

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Step 2.2.1

Divide each term in $- 5 x > 23$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 x}{- 5} < \frac{23}{- 5}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 5$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} < \frac{23}{- 5}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{23}{- 5}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Move the negative in front of the fraction.

$x < - \frac{23}{5}$

Step 3

Solve $5 x - 10 > 17$ for $x$ .

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Step 3.1

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

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Step 3.1.1

Add $10$ to both sides of the inequality.

$5 x > 17 + 10$

Step 3.1.2

Add $17$ and $10$ .

$5 x > 27$

Step 3.2

Divide each term in $5 x > 27$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 x > 27$ by $5$ .

$\frac{5 x}{5} > \frac{27}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} > \frac{27}{5}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{27}{5}$

Step 4

Find the union of the solutions.

$x < - \frac{23}{5}$ or $x > \frac{27}{5}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{23}{5} or x > \frac{27}{5}$

Interval Notation:

$(- \infty, - \frac{23}{5}) \cup (\frac{27}{5}, \infty)$

Step 6