Finite Math Examples

$(7) | x + 5 | + 8 > 5$

Step 1

Write $7 | x + 5 | + 8 > 5$ 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.

$x + 5 \geq 0$

Step 1.2

Subtract $5$ from both sides of the inequality.

$x \geq - 5$

Step 1.3

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

$7 (x + 5) + 8 > 5$

Step 1.4

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

$x + 5 < 0$

Step 1.5

Subtract $5$ from both sides of the inequality.

$x < - 5$

Step 1.6

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

$7 (- (x + 5)) + 8 > 5$

Step 1.7

Write as a piecewise.

${\begin{cases} 7 (x + 5) + 8 > 5 & x \geq - 5 \\ 7 (- (x + 5)) + 8 > 5 & x < - 5 \end{cases}$

Step 1.8

Simplify $7 (x + 5) + 8 > 5$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Apply the distributive property.

${\begin{cases} 7 x + 7 \cdot 5 + 8 > 5 & x \geq - 5 \\ 7 (- (x + 5)) + 8 > 5 & x < - 5 \end{cases}$

Step 1.8.1.2

Multiply $7$ by $5$ .

${\begin{cases} 7 x + 35 + 8 > 5 & x \geq - 5 \\ 7 (- (x + 5)) + 8 > 5 & x < - 5 \end{cases}$

Step 1.8.2

Add $35$ and $8$ .

${\begin{cases} 7 x + 43 > 5 & x \geq - 5 \\ 7 (- (x + 5)) + 8 > 5 & x < - 5 \end{cases}$

Step 1.9

Simplify $7 (- (x + 5)) + 8 > 5$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 7 x + 43 > 5 & x \geq - 5 \\ 7 (- x - 1 \cdot 5) + 8 > 5 & x < - 5 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $5$ .

${\begin{cases} 7 x + 43 > 5 & x \geq - 5 \\ 7 (- x - 5) + 8 > 5 & x < - 5 \end{cases}$

Step 1.9.1.3

Apply the distributive property.

${\begin{cases} 7 x + 43 > 5 & x \geq - 5 \\ 7 (- x) + 7 \cdot - 5 + 8 > 5 & x < - 5 \end{cases}$

Step 1.9.1.4

Multiply $- 1$ by $7$ .

${\begin{cases} 7 x + 43 > 5 & x \geq - 5 \\ - 7 x + 7 \cdot - 5 + 8 > 5 & x < - 5 \end{cases}$

Step 1.9.1.5

Multiply $7$ by $- 5$ .

${\begin{cases} 7 x + 43 > 5 & x \geq - 5 \\ - 7 x - 35 + 8 > 5 & x < - 5 \end{cases}$

Step 1.9.2

Add $- 35$ and $8$ .

${\begin{cases} 7 x + 43 > 5 & x \geq - 5 \\ - 7 x - 27 > 5 & x < - 5 \end{cases}$

Step 2

Solve $7 x + 43 > 5$ when $x \geq - 5$ .

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Step 2.1

Solve $7 x + 43 > 5$ for $x$ .

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Step 2.1.1

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

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Step 2.1.1.1

Subtract $43$ from both sides of the inequality.

$7 x > 5 - 43$

Step 2.1.1.2

Subtract $43$ from $5$ .

$7 x > - 38$

Step 2.1.2

Divide each term in $7 x > - 38$ by $7$ and simplify.

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Step 2.1.2.1

Divide each term in $7 x > - 38$ by $7$ .

$\frac{7 x}{7} > \frac{- 38}{7}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $7$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} > \frac{- 38}{7}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 38}{7}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Move the negative in front of the fraction.

$x > - \frac{38}{7}$

Step 2.2

Find the intersection of $x > - \frac{38}{7}$ and $x \geq - 5$ .

$x \geq - 5$

Step 3

Solve $- 7 x - 27 > 5$ when $x < - 5$ .

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Step 3.1

Solve $- 7 x - 27 > 5$ for $x$ .

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Step 3.1.1

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

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Step 3.1.1.1

Add $27$ to both sides of the inequality.

$- 7 x > 5 + 27$

Step 3.1.1.2

Add $5$ and $27$ .

$- 7 x > 32$

Step 3.1.2

Divide each term in $- 7 x > 32$ by $- 7$ and simplify.

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Step 3.1.2.1

Divide each term in $- 7 x > 32$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} < \frac{32}{- 7}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 7$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} < \frac{32}{- 7}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{32}{- 7}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Move the negative in front of the fraction.

$x < - \frac{32}{7}$

Step 3.2

Find the intersection of $x < - \frac{32}{7}$ and $x < - 5$ .

$x < - 5$

Step 4

Find the union of the solutions.

All real numbers

Step 5

Convert the inequality to interval notation.

$(- \infty, \infty)$

Step 6