Calculus Examples

$| \frac{7 x + 28}{4} | > 7$

Step 1

Write $| \frac{7 x + 28}{4} | > 7$ 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.

$\frac{7 x + 28}{4} \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Multiply both sides by $4$ .

$\frac{7 x + 28}{4} \cdot 4 \geq 0 \cdot 4$

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

Simplify $\frac{7 x + 28}{4} \cdot 4$ .

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Step 1.2.2.1.1.1

Factor $7$ out of $7 x + 28$ .

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Step 1.2.2.1.1.1.1

Factor $7$ out of $7 x$ .

$\frac{7 (x) + 28}{4} \cdot 4 \geq 0 \cdot 4$

Step 1.2.2.1.1.1.2

Factor $7$ out of $28$ .

$\frac{7 x + 7 \cdot 4}{4} \cdot 4 \geq 0 \cdot 4$

Step 1.2.2.1.1.1.3

Factor $7$ out of $7 x + 7 \cdot 4$ .

$\frac{7 (x + 4)}{4} \cdot 4 \geq 0 \cdot 4$

Step 1.2.2.1.1.2

Cancel the common factor of $4$ .

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Step 1.2.2.1.1.2.1

Cancel the common factor.

$\frac{7 (x + 4)}{4} \cdot 4 \geq 0 \cdot 4$

Step 1.2.2.1.1.2.2

Rewrite the expression.

$7 (x + 4) \geq 0 \cdot 4$

Step 1.2.2.1.1.3

Apply the distributive property.

$7 x + 7 \cdot 4 \geq 0 \cdot 4$

Step 1.2.2.1.1.4

Multiply $7$ by $4$ .

$7 x + 28 \geq 0 \cdot 4$

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Multiply $0$ by $4$ .

$7 x + 28 \geq 0$

Step 1.2.3

Solve for $x$ .

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Step 1.2.3.1

Subtract $28$ from both sides of the inequality.

$7 x \geq - 28$

Step 1.2.3.2

Divide each term in $7 x \geq - 28$ by $7$ and simplify.

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Step 1.2.3.2.1

Divide each term in $7 x \geq - 28$ by $7$ .

$\frac{7 x}{7} \geq \frac{- 28}{7}$

Step 1.2.3.2.2

Simplify the left side.

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Step 1.2.3.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.3.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} \geq \frac{- 28}{7}$

Step 1.2.3.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 28}{7}$

Step 1.2.3.2.3

Simplify the right side.

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Step 1.2.3.2.3.1

Divide $- 28$ by $7$ .

$x \geq - 4$

Step 1.3

In the piece where $\frac{7 x + 28}{4}$ is non-negative, remove the absolute value.

$\frac{7 x + 28}{4} > 7$

Step 1.4

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

$\frac{7 x + 28}{4} < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Multiply both sides by $4$ .

$\frac{7 x + 28}{4} \cdot 4 < 0 \cdot 4$

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

Simplify $\frac{7 x + 28}{4} \cdot 4$ .

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Step 1.5.2.1.1.1

Factor $7$ out of $7 x + 28$ .

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Step 1.5.2.1.1.1.1

Factor $7$ out of $7 x$ .

$\frac{7 (x) + 28}{4} \cdot 4 < 0 \cdot 4$

Step 1.5.2.1.1.1.2

Factor $7$ out of $28$ .

$\frac{7 x + 7 \cdot 4}{4} \cdot 4 < 0 \cdot 4$

Step 1.5.2.1.1.1.3

Factor $7$ out of $7 x + 7 \cdot 4$ .

$\frac{7 (x + 4)}{4} \cdot 4 < 0 \cdot 4$

Step 1.5.2.1.1.2

Cancel the common factor of $4$ .

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Step 1.5.2.1.1.2.1

Cancel the common factor.

$\frac{7 (x + 4)}{4} \cdot 4 < 0 \cdot 4$

Step 1.5.2.1.1.2.2

Rewrite the expression.

$7 (x + 4) < 0 \cdot 4$

Step 1.5.2.1.1.3

Apply the distributive property.

$7 x + 7 \cdot 4 < 0 \cdot 4$

Step 1.5.2.1.1.4

Multiply $7$ by $4$ .

$7 x + 28 < 0 \cdot 4$

Step 1.5.2.2

Simplify the right side.

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Step 1.5.2.2.1

Multiply $0$ by $4$ .

$7 x + 28 < 0$

Step 1.5.3

Solve for $x$ .

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Step 1.5.3.1

Subtract $28$ from both sides of the inequality.

$7 x < - 28$

Step 1.5.3.2

Divide each term in $7 x < - 28$ by $7$ and simplify.

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Step 1.5.3.2.1

Divide each term in $7 x < - 28$ by $7$ .

$\frac{7 x}{7} < \frac{- 28}{7}$

Step 1.5.3.2.2

Simplify the left side.

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Step 1.5.3.2.2.1

Cancel the common factor of $7$ .

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Step 1.5.3.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} < \frac{- 28}{7}$

Step 1.5.3.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 28}{7}$

Step 1.5.3.2.3

Simplify the right side.

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Step 1.5.3.2.3.1

Divide $- 28$ by $7$ .

$x < - 4$

Step 1.6

In the piece where $\frac{7 x + 28}{4}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{7 x + 28}{4} > 7$

Step 1.7

Write as a piecewise.

${\begin{cases} \frac{7 x + 28}{4} > 7 & x \geq - 4 \\ - \frac{7 x + 28}{4} > 7 & x < - 4 \end{cases}$

Step 1.8

Factor $7$ out of $7 x + 28$ .

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Step 1.8.1

Factor $7$ out of $7 x$ .

${\begin{cases} \frac{7 (x) + 28}{4} > 7 & x \geq - 4 \\ - \frac{7 x + 28}{4} > 7 & x < - 4 \end{cases}$

Step 1.8.2

Factor $7$ out of $28$ .

${\begin{cases} \frac{7 x + 7 \cdot 4}{4} > 7 & x \geq - 4 \\ - \frac{7 x + 28}{4} > 7 & x < - 4 \end{cases}$

Step 1.8.3

Factor $7$ out of $7 x + 7 \cdot 4$ .

${\begin{cases} \frac{7 (x + 4)}{4} > 7 & x \geq - 4 \\ - \frac{7 x + 28}{4} > 7 & x < - 4 \end{cases}$

Step 1.9

Factor $7$ out of $7 x + 28$ .

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Step 1.9.1

Factor $7$ out of $7 x$ .

${\begin{cases} \frac{7 (x + 4)}{4} > 7 & x \geq - 4 \\ - \frac{7 (x) + 28}{4} > 7 & x < - 4 \end{cases}$

Step 1.9.2

Factor $7$ out of $28$ .

${\begin{cases} \frac{7 (x + 4)}{4} > 7 & x \geq - 4 \\ - \frac{7 x + 7 \cdot 4}{4} > 7 & x < - 4 \end{cases}$

Step 1.9.3

Factor $7$ out of $7 x + 7 \cdot 4$ .

${\begin{cases} \frac{7 (x + 4)}{4} > 7 & x \geq - 4 \\ - \frac{7 (x + 4)}{4} > 7 & x < - 4 \end{cases}$

Step 2

Solve $\frac{7 (x + 4)}{4} > 7$ for $x$ .

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Step 2.1

Multiply both sides by $4$ .

$\frac{7 (x + 4)}{4} \cdot 4 > 7 \cdot 4$

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

Simplify $\frac{7 (x + 4)}{4} \cdot 4$ .

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Step 2.2.1.1.1

Cancel the common factor of $4$ .

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Step 2.2.1.1.1.1

Cancel the common factor.

$\frac{7 (x + 4)}{4} \cdot 4 > 7 \cdot 4$

Step 2.2.1.1.1.2

Rewrite the expression.

$7 (x + 4) > 7 \cdot 4$

Step 2.2.1.1.2

Apply the distributive property.

$7 x + 7 \cdot 4 > 7 \cdot 4$

Step 2.2.1.1.3

Multiply $7$ by $4$ .

$7 x + 28 > 7 \cdot 4$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Multiply $7$ by $4$ .

$7 x + 28 > 28$

Step 2.3

Solve for $x$ .

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Step 2.3.1

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

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Step 2.3.1.1

Subtract $28$ from both sides of the inequality.

$7 x > 28 - 28$

Step 2.3.1.2

Subtract $28$ from $28$ .

$7 x > 0$

Step 2.3.2

Divide each term in $7 x > 0$ by $7$ and simplify.

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Step 2.3.2.1

Divide each term in $7 x > 0$ by $7$ .

$\frac{7 x}{7} > \frac{0}{7}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $7$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} > \frac{0}{7}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{0}{7}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $0$ by $7$ .

$x > 0$

Step 3

Solve $- \frac{7 (x + 4)}{4} > 7$ for $x$ .

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Step 3.1

Divide each term in $- \frac{7 (x + 4)}{4} > 7$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- \frac{7 (x + 4)}{4} > 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{7 (x + 4)}{4}}{- 1} < \frac{7}{- 1}$

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.

$\frac{\frac{7 (x + 4)}{4}}{1} < \frac{7}{- 1}$

Step 3.1.2.2

Divide $\frac{7 (x + 4)}{4}$ by $1$ .

$\frac{7 (x + 4)}{4} < \frac{7}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $7$ by $- 1$ .

$\frac{7 (x + 4)}{4} < - 7$

Step 3.2

Multiply both sides by $4$ .

$\frac{7 (x + 4)}{4} \cdot 4 < - 7 \cdot 4$

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

Simplify $\frac{7 (x + 4)}{4} \cdot 4$ .

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Step 3.3.1.1.1

Cancel the common factor of $4$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{7 (x + 4)}{4} \cdot 4 < - 7 \cdot 4$

Step 3.3.1.1.1.2

Rewrite the expression.

$7 (x + 4) < - 7 \cdot 4$

Step 3.3.1.1.2

Apply the distributive property.

$7 x + 7 \cdot 4 < - 7 \cdot 4$

Step 3.3.1.1.3

Multiply $7$ by $4$ .

$7 x + 28 < - 7 \cdot 4$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply $- 7$ by $4$ .

$7 x + 28 < - 28$

Step 3.4

Solve for $x$ .

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Step 3.4.1

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

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Step 3.4.1.1

Subtract $28$ from both sides of the inequality.

$7 x < - 28 - 28$

Step 3.4.1.2

Subtract $28$ from $- 28$ .

$7 x < - 56$

Step 3.4.2

Divide each term in $7 x < - 56$ by $7$ and simplify.

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Step 3.4.2.1

Divide each term in $7 x < - 56$ by $7$ .

$\frac{7 x}{7} < \frac{- 56}{7}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $7$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} < \frac{- 56}{7}$

Step 3.4.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 56}{7}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $- 56$ by $7$ .

$x < - 8$

Step 4

Find the union of the solutions.

$x < - 8$ or $x > 0$

Step 5

Convert the inequality to interval notation.

$(- \infty, - 8) \cup (0, \infty)$

Step 6