Finite Math Examples

$| \frac{5}{7} x + 5 | > 10$

Step 1

Write $| \frac{5}{7} x + 5 | > 10$ 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.

$\frac{5}{7} x + 5 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Combine $\frac{5}{7}$ and $x$ .

$\frac{5 x}{7} + 5 \geq 0$

Step 1.2.2

Subtract $5$ from both sides of the inequality.

$\frac{5 x}{7} \geq - 5$

Step 1.2.3

Multiply both sides by $7$ .

$\frac{5 x}{7} \cdot 7 \geq - 5 \cdot 7$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the left side.

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Step 1.2.4.1.1

Cancel the common factor of $7$ .

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Step 1.2.4.1.1.1

Cancel the common factor.

$\frac{5 x}{7} \cdot 7 \geq - 5 \cdot 7$

Step 1.2.4.1.1.2

Rewrite the expression.

$5 x \geq - 5 \cdot 7$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply $- 5$ by $7$ .

$5 x \geq - 35$

Step 1.2.5

Divide each term in $5 x \geq - 35$ by $5$ and simplify.

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Step 1.2.5.1

Divide each term in $5 x \geq - 35$ by $5$ .

$\frac{5 x}{5} \geq \frac{- 35}{5}$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Cancel the common factor of $5$ .

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Step 1.2.5.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \geq \frac{- 35}{5}$

Step 1.2.5.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 35}{5}$

Step 1.2.5.3

Simplify the right side.

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Step 1.2.5.3.1

Divide $- 35$ by $5$ .

$x \geq - 7$

Step 1.3

In the piece where $\frac{5}{7} x + 5$ is non-negative, remove the absolute value.

$\frac{5}{7} x + 5 > 10$

Step 1.4

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

$\frac{5}{7} x + 5 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Combine $\frac{5}{7}$ and $x$ .

$\frac{5 x}{7} + 5 < 0$

Step 1.5.2

Subtract $5$ from both sides of the inequality.

$\frac{5 x}{7} < - 5$

Step 1.5.3

Multiply both sides by $7$ .

$\frac{5 x}{7} \cdot 7 < - 5 \cdot 7$

Step 1.5.4

Simplify.

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Step 1.5.4.1

Simplify the left side.

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Step 1.5.4.1.1

Cancel the common factor of $7$ .

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Step 1.5.4.1.1.1

Cancel the common factor.

$\frac{5 x}{7} \cdot 7 < - 5 \cdot 7$

Step 1.5.4.1.1.2

Rewrite the expression.

$5 x < - 5 \cdot 7$

Step 1.5.4.2

Simplify the right side.

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Step 1.5.4.2.1

Multiply $- 5$ by $7$ .

$5 x < - 35$

Step 1.5.5

Divide each term in $5 x < - 35$ by $5$ and simplify.

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Step 1.5.5.1

Divide each term in $5 x < - 35$ by $5$ .

$\frac{5 x}{5} < \frac{- 35}{5}$

Step 1.5.5.2

Simplify the left side.

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Step 1.5.5.2.1

Cancel the common factor of $5$ .

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Step 1.5.5.2.1.1

Cancel the common factor.

$\frac{5 x}{5} < \frac{- 35}{5}$

Step 1.5.5.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 35}{5}$

Step 1.5.5.3

Simplify the right side.

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Step 1.5.5.3.1

Divide $- 35$ by $5$ .

$x < - 7$

Step 1.6

In the piece where $\frac{5}{7} x + 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (\frac{5}{7} x + 5) > 10$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{5}{7} x + 5 > 10 & x \geq - 7 \\ - (\frac{5}{7} x + 5) > 10 & x < - 7 \end{cases}$

Step 1.8

Combine $\frac{5}{7}$ and $x$ .

${\begin{cases} \frac{5 x}{7} + 5 > 10 & x \geq - 7 \\ - (\frac{5}{7} x + 5) > 10 & x < - 7 \end{cases}$

Step 1.9

Simplify $- (\frac{5}{7} x + 5) > 10$ .

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Step 1.9.1

Combine $\frac{5}{7}$ and $x$ .

${\begin{cases} \frac{5 x}{7} + 5 > 10 & x \geq - 7 \\ - (\frac{5 x}{7} + 5) > 10 & x < - 7 \end{cases}$

Step 1.9.2

Apply the distributive property.

${\begin{cases} \frac{5 x}{7} + 5 > 10 & x \geq - 7 \\ - \frac{5 x}{7} - 1 \cdot 5 > 10 & x < - 7 \end{cases}$

Step 1.9.3

Multiply $- 1$ by $5$ .

${\begin{cases} \frac{5 x}{7} + 5 > 10 & x \geq - 7 \\ - \frac{5 x}{7} - 5 > 10 & x < - 7 \end{cases}$

Step 2

Solve $\frac{5 x}{7} + 5 > 10$ 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

Subtract $5$ from both sides of the inequality.

$\frac{5 x}{7} > 10 - 5$

Step 2.1.2

Subtract $5$ from $10$ .

$\frac{5 x}{7} > 5$

Step 2.2

Multiply both sides by $7$ .

$\frac{5 x}{7} \cdot 7 > 5 \cdot 7$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Cancel the common factor of $7$ .

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Step 2.3.1.1.1

Cancel the common factor.

$\frac{5 x}{7} \cdot 7 > 5 \cdot 7$

Step 2.3.1.1.2

Rewrite the expression.

$5 x > 5 \cdot 7$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $5$ by $7$ .

$5 x > 35$

Step 2.4

Divide each term in $5 x > 35$ by $5$ and simplify.

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Step 2.4.1

Divide each term in $5 x > 35$ by $5$ .

$\frac{5 x}{5} > \frac{35}{5}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $5$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} > \frac{35}{5}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x > \frac{35}{5}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Divide $35$ by $5$ .

$x > 7$

Step 3

Solve $- \frac{5 x}{7} - 5 > 10$ 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 $5$ to both sides of the inequality.

$- \frac{5 x}{7} > 10 + 5$

Step 3.1.2

Add $10$ and $5$ .

$- \frac{5 x}{7} > 15$

Step 3.2

Divide each term in $- \frac{5 x}{7} > 15$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- \frac{5 x}{7} > 15$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{5 x}{7}}{- 1} < \frac{15}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{5 x}{7}}{1} < \frac{15}{- 1}$

Step 3.2.2.2

Divide $\frac{5 x}{7}$ by $1$ .

$\frac{5 x}{7} < \frac{15}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $15$ by $- 1$ .

$\frac{5 x}{7} < - 15$

Step 3.3

Multiply both sides by $7$ .

$\frac{5 x}{7} \cdot 7 < - 15 \cdot 7$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Cancel the common factor of $7$ .

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Step 3.4.1.1.1

Cancel the common factor.

$\frac{5 x}{7} \cdot 7 < - 15 \cdot 7$

Step 3.4.1.1.2

Rewrite the expression.

$5 x < - 15 \cdot 7$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Multiply $- 15$ by $7$ .

$5 x < - 105$

Step 3.5

Divide each term in $5 x < - 105$ by $5$ and simplify.

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Step 3.5.1

Divide each term in $5 x < - 105$ by $5$ .

$\frac{5 x}{5} < \frac{- 105}{5}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $5$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{5 x}{5} < \frac{- 105}{5}$

Step 3.5.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 105}{5}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Divide $- 105$ by $5$ .

$x < - 21$

Step 4

Find the union of the solutions.

$x < - 21$ or $x > 7$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - 21 or x > 7$

Interval Notation:

$(- \infty, - 21) \cup (7, \infty)$

Step 6