Algebra Examples

$2 > - | \frac{x - 8}{5} + \frac{3}{5} |$

Step 1

Rewrite so $x$ is on the left side of the inequality.

$- | \frac{x - 8}{5} + \frac{3}{5} | < 2$

Step 2

Divide each term in $- | \frac{x - 8}{5} + \frac{3}{5} | < 2$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- | \frac{x - 8}{5} + \frac{3}{5} | < 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- | \frac{x - 8}{5} + \frac{3}{5} |}{- 1} > \frac{2}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{| \frac{x - 8}{5} + \frac{3}{5} |}{1} > \frac{2}{- 1}$

Step 2.2.2

Divide $| \frac{x - 8}{5} + \frac{3}{5} |$ by $1$ .

$| \frac{x - 8}{5} + \frac{3}{5} | > \frac{2}{- 1}$

Step 2.2.3

Combine the numerators over the common denominator.

$| \frac{x - 8 + 3}{5} | > \frac{2}{- 1}$

Step 2.2.4

Add $- 8$ and $3$ .

$| \frac{x - 5}{5} | > \frac{2}{- 1}$

Step 2.2.5

Remove non-negative terms from the absolute value.

$\frac{| x - 5 |}{5} > \frac{2}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $2$ by $- 1$ .

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

Step 3

Write $\frac{| x - 5 |}{5} > - 2$ as a piecewise.

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Step 3.1

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

$x - 5 \geq 0$

Step 3.2

Add $5$ to both sides of the inequality.

$x \geq 5$

Step 3.3

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

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

Step 3.4

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

$x - 5 < 0$

Step 3.5

Add $5$ to both sides of the inequality.

$x < 5$

Step 3.6

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

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

Step 3.7

Write as a piecewise.

${\begin{cases} \frac{x - 5}{5} > - 2 & x \geq 5 \\ \frac{- (x - 5)}{5} > - 2 & x < 5 \end{cases}$

Step 3.8

Move the negative in front of the fraction.

${\begin{cases} \frac{x - 5}{5} > - 2 & x \geq 5 \\ - \frac{x - 5}{5} > - 2 & x < 5 \end{cases}$

Step 4

Solve $\frac{x - 5}{5} > - 2$ when $x \geq 5$ .

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Step 4.1

Solve $\frac{x - 5}{5} > - 2$ for $x$ .

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Step 4.1.1

Multiply both sides by $5$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the left side.

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Step 4.1.2.1.1

Cancel the common factor of $5$ .

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Step 4.1.2.1.1.1

Cancel the common factor.

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

Step 4.1.2.1.1.2

Rewrite the expression.

$x - 5 > - 2 \cdot 5$

Step 4.1.2.2

Simplify the right side.

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Step 4.1.2.2.1

Multiply $- 2$ by $5$ .

$x - 5 > - 10$

Step 4.1.3

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

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Step 4.1.3.1

Add $5$ to both sides of the inequality.

$x > - 10 + 5$

Step 4.1.3.2

Add $- 10$ and $5$ .

$x > - 5$

Step 4.2

Find the intersection of $x > - 5$ and $x \geq 5$ .

$x \geq 5$

Step 5

Solve $- \frac{x - 5}{5} > - 2$ when $x < 5$ .

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Step 5.1

Solve $- \frac{x - 5}{5} > - 2$ for $x$ .

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Step 5.1.1

Divide each term in $- \frac{x - 5}{5} > - 2$ by $- 1$ and simplify.

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Step 5.1.1.1

Divide each term in $- \frac{x - 5}{5} > - 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x - 5}{5}}{- 1} < \frac{- 2}{- 1}$

Step 5.1.1.2

Simplify the left side.

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Step 5.1.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x - 5}{5}}{1} < \frac{- 2}{- 1}$

Step 5.1.1.2.2

Divide $\frac{x - 5}{5}$ by $1$ .

$\frac{x - 5}{5} < \frac{- 2}{- 1}$

Step 5.1.1.3

Simplify the right side.

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Step 5.1.1.3.1

Divide $- 2$ by $- 1$ .

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

Step 5.1.2

Multiply both sides by $5$ .

$\frac{x - 5}{5} \cdot 5 < 2 \cdot 5$

Step 5.1.3

Simplify.

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Step 5.1.3.1

Simplify the left side.

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Step 5.1.3.1.1

Cancel the common factor of $5$ .

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Step 5.1.3.1.1.1

Cancel the common factor.

$\frac{x - 5}{5} \cdot 5 < 2 \cdot 5$

Step 5.1.3.1.1.2

Rewrite the expression.

$x - 5 < 2 \cdot 5$

Step 5.1.3.2

Simplify the right side.

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Step 5.1.3.2.1

Multiply $2$ by $5$ .

$x - 5 < 10$

Step 5.1.4

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

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Step 5.1.4.1

Add $5$ to both sides of the inequality.

$x < 10 + 5$

Step 5.1.4.2

Add $10$ and $5$ .

$x < 15$

Step 5.2

Find the intersection of $x < 15$ and $x < 5$ .

$x < 5$

Step 6

Find the union of the solutions for any value of $x$ .

All real numbers

Step 7

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Step 8