Pre-Algebra Examples

$\frac{6}{14 (x - 7)} > - \frac{8}{14 (x + 7)}$

Step 1

Cancel the common factor of $6$ and $14$ .

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Step 1.1

Factor $2$ out of $6$ .

$\frac{2 (3)}{14 (x - 7)} > - \frac{8}{14 (x + 7)}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $14 (x - 7)$ .

$\frac{2 (3)}{2 (7 (x - 7))} > - \frac{8}{14 (x + 7)}$

Step 1.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 (7 (x - 7))} > - \frac{8}{14 (x + 7)}$

Step 1.2.3

Rewrite the expression.

$\frac{3}{7 (x - 7)} > - \frac{8}{14 (x + 7)}$

Step 2

Multiply both sides by $7 (x - 7)$ .

$\frac{3}{7 (x - 7)} (7 (x - 7)) = - \frac{8}{14 (x + 7)} (7 (x - 7))$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{3}{7 (x - 7)} (7 (x - 7))$ .

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Step 3.1.1.1

Rewrite using the commutative property of multiplication.

$7 \frac{3}{7 (x - 7)} (x - 7) = - \frac{8}{14 (x + 7)} (7 (x - 7))$

Step 3.1.1.2

Cancel the common factor of $7$ .

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Step 3.1.1.2.1

Cancel the common factor.

$7 \frac{3}{7 (x - 7)} (x - 7) = - \frac{8}{14 (x + 7)} (7 (x - 7))$

Step 3.1.1.2.2

Rewrite the expression.

$\frac{3}{x - 7} (x - 7) = - \frac{8}{14 (x + 7)} (7 (x - 7))$

Step 3.1.1.3

Cancel the common factor of $x - 7$ .

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Step 3.1.1.3.1

Cancel the common factor.

$\frac{3}{x - 7} (x - 7) = - \frac{8}{14 (x + 7)} (7 (x - 7))$

Step 3.1.1.3.2

Rewrite the expression.

$3 = - \frac{8}{14 (x + 7)} (7 (x - 7))$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- \frac{8}{14 (x + 7)} (7 (x - 7))$ .

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Step 3.2.1.1

Cancel the common factor of $7$ .

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Step 3.2.1.1.1

Move the leading negative in $- \frac{8}{14 (x + 7)}$ into the numerator.

$3 = \frac{- 8}{14 (x + 7)} (7 (x - 7))$

Step 3.2.1.1.2

Factor $7$ out of $14 (x + 7)$ .

$3 = \frac{- 8}{7 (2 (x + 7))} (7 (x - 7))$

Step 3.2.1.1.3

Cancel the common factor.

$3 = \frac{- 8}{7 (2 (x + 7))} (7 (x - 7))$

Step 3.2.1.1.4

Rewrite the expression.

$3 = \frac{- 8}{2 (x + 7)} (x - 7)$

Step 3.2.1.2

Cancel the common factor of $- 8$ and $2$ .

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Step 3.2.1.2.1

Factor $2$ out of $- 8$ .

$3 = \frac{2 \cdot - 4}{2 (x + 7)} (x - 7)$

Step 3.2.1.2.2

Cancel the common factors.

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Step 3.2.1.2.2.1

Cancel the common factor.

$3 = \frac{2 \cdot - 4}{2 (x + 7)} (x - 7)$

Step 3.2.1.2.2.2

Rewrite the expression.

$3 = \frac{- 4}{x + 7} (x - 7)$

Step 3.2.1.3

Move the negative in front of the fraction.

$3 = - \frac{4}{x + 7} (x - 7)$

Step 3.2.1.4

Apply the distributive property.

$3 = - \frac{4}{x + 7} x - \frac{4}{x + 7} \cdot - 7$

Step 3.2.1.5

Combine $x$ and $\frac{4}{x + 7}$ .

$3 = - \frac{x \cdot 4}{x + 7} - \frac{4}{x + 7} \cdot - 7$

Step 3.2.1.6

Multiply $- \frac{4}{x + 7} \cdot - 7$ .

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Step 3.2.1.6.1

Multiply $- 7$ by $- 1$ .

$3 = - \frac{x \cdot 4}{x + 7} + 7 \frac{4}{x + 7}$

Step 3.2.1.6.2

Combine $7$ and $\frac{4}{x + 7}$ .

$3 = - \frac{x \cdot 4}{x + 7} + \frac{7 \cdot 4}{x + 7}$

Step 3.2.1.6.3

Multiply $7$ by $4$ .

$3 = - \frac{x \cdot 4}{x + 7} + \frac{28}{x + 7}$

Step 3.2.1.7

Combine the numerators over the common denominator.

$3 = \frac{- x \cdot 4 + 28}{x + 7}$

Step 3.2.1.8

Multiply $4$ by $- 1$ .

$3 = \frac{- 4 x + 28}{x + 7}$

Step 3.2.1.9

Factor $4$ out of $- 4 x + 28$ .

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Step 3.2.1.9.1

Factor $4$ out of $- 4 x$ .

$3 = \frac{4 (- x) + 28}{x + 7}$

Step 3.2.1.9.2

Factor $4$ out of $28$ .

$3 = \frac{4 (- x) + 4 (7)}{x + 7}$

Step 3.2.1.9.3

Factor $4$ out of $4 (- x) + 4 (7)$ .

$3 = \frac{4 (- x + 7)}{x + 7}$

Step 3.2.1.10

Factor $- 1$ out of $- x$ .

$3 = \frac{4 (- (x) + 7)}{x + 7}$

Step 3.2.1.11

Rewrite $7$ as $- 1 (- 7)$ .

$3 = \frac{4 (- (x) - 1 (- 7))}{x + 7}$

Step 3.2.1.12

Factor $- 1$ out of $- (x) - 1 (- 7)$ .

$3 = \frac{4 (- (x - 7))}{x + 7}$

Step 3.2.1.13

Simplify the expression.

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Step 3.2.1.13.1

Rewrite $- (x - 7)$ as $- 1 (x - 7)$ .

$3 = \frac{4 (- 1 (x - 7))}{x + 7}$

Step 3.2.1.13.2

Move the negative in front of the fraction.

$3 = - \frac{4 (x - 7)}{x + 7}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $- \frac{4 (x - 7)}{x + 7} = 3$ .

$- \frac{4 (x - 7)}{x + 7} = 3$

Step 4.2

Find the LCD of the terms in the equation.

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Step 4.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x + 7, 1$

Step 4.2.2

Remove parentheses.

$x + 7, 1$

Step 4.2.3

The LCM of one and any expression is the expression.

$x + 7$

Step 4.3

Multiply each term in $- \frac{4 (x - 7)}{x + 7} = 3$ by $x + 7$ to eliminate the fractions.

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Step 4.3.1

Multiply each term in $- \frac{4 (x - 7)}{x + 7} = 3$ by $x + 7$ .

$- \frac{4 (x - 7)}{x + 7} (x + 7) = 3 (x + 7)$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $x + 7$ .

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Step 4.3.2.1.1

Move the leading negative in $- \frac{4 (x - 7)}{x + 7}$ into the numerator.

$\frac{- 4 (x - 7)}{x + 7} (x + 7) = 3 (x + 7)$

Step 4.3.2.1.2

Cancel the common factor.

$\frac{- 4 (x - 7)}{x + 7} (x + 7) = 3 (x + 7)$

Step 4.3.2.1.3

Rewrite the expression.

$- 4 (x - 7) = 3 (x + 7)$

Step 4.3.2.2

Apply the distributive property.

$- 4 x - 4 \cdot - 7 = 3 (x + 7)$

Step 4.3.2.3

Multiply $- 4$ by $- 7$ .

$- 4 x + 28 = 3 (x + 7)$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Apply the distributive property.

$- 4 x + 28 = 3 x + 3 \cdot 7$

Step 4.3.3.2

Multiply $3$ by $7$ .

$- 4 x + 28 = 3 x + 21$

Step 4.4

Solve the equation.

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Step 4.4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.4.1.1

Subtract $3 x$ from both sides of the equation.

$- 4 x + 28 - 3 x = 21$

Step 4.4.1.2

Subtract $3 x$ from $- 4 x$ .

$- 7 x + 28 = 21$

Step 4.4.2

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

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Step 4.4.2.1

Subtract $28$ from both sides of the equation.

$- 7 x = 21 - 28$

Step 4.4.2.2

Subtract $28$ from $21$ .

$- 7 x = - 7$

Step 4.4.3

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

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Step 4.4.3.1

Divide each term in $- 7 x = - 7$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{- 7}{- 7}$

Step 4.4.3.2

Simplify the left side.

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Step 4.4.3.2.1

Cancel the common factor of $- 7$ .

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Step 4.4.3.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{- 7}{- 7}$

Step 4.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{- 7}$

Step 4.4.3.3

Simplify the right side.

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Step 4.4.3.3.1

Divide $- 7$ by $- 7$ .

$x = 1$

Step 5

Find the domain of $\frac{3}{7 (x - 7)} + \frac{4}{7 (x + 7)}$ .

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Step 5.1

Set the denominator in $\frac{3}{7 (x - 7)}$ equal to $0$ to find where the expression is undefined.

$7 (x - 7) = 0$

Step 5.2

Solve for $x$ .

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Step 5.2.1

Divide each term in $7 (x - 7) = 0$ by $7$ and simplify.

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Step 5.2.1.1

Divide each term in $7 (x - 7) = 0$ by $7$ .

$\frac{7 (x - 7)}{7} = \frac{0}{7}$

Step 5.2.1.2

Simplify the left side.

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Step 5.2.1.2.1

Cancel the common factor of $7$ .

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Step 5.2.1.2.1.1

Cancel the common factor.

$\frac{7 (x - 7)}{7} = \frac{0}{7}$

Step 5.2.1.2.1.2

Divide $x - 7$ by $1$ .

$x - 7 = \frac{0}{7}$

Step 5.2.1.3

Simplify the right side.

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Step 5.2.1.3.1

Divide $0$ by $7$ .

$x - 7 = 0$

Step 5.2.2

Add $7$ to both sides of the equation.

$x = 7$

Step 5.3

Set the denominator in $\frac{4}{7 (x + 7)}$ equal to $0$ to find where the expression is undefined.

$7 (x + 7) = 0$

Step 5.4

Solve for $x$ .

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Step 5.4.1

Divide each term in $7 (x + 7) = 0$ by $7$ and simplify.

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Step 5.4.1.1

Divide each term in $7 (x + 7) = 0$ by $7$ .

$\frac{7 (x + 7)}{7} = \frac{0}{7}$

Step 5.4.1.2

Simplify the left side.

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Step 5.4.1.2.1

Cancel the common factor of $7$ .

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Step 5.4.1.2.1.1

Cancel the common factor.

$\frac{7 (x + 7)}{7} = \frac{0}{7}$

Step 5.4.1.2.1.2

Divide $x + 7$ by $1$ .

$x + 7 = \frac{0}{7}$

Step 5.4.1.3

Simplify the right side.

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Step 5.4.1.3.1

Divide $0$ by $7$ .

$x + 7 = 0$

Step 5.4.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 5.5

The domain is all values of $x$ that make the expression defined.

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

Step 6

Use each root to create test intervals.

$x < - 7$

$- 7 < x < 1$

$1 < x < 7$

$x > 7$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 7.1

Test a value on the interval $x < - 7$ to see if it makes the inequality true.

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Step 7.1.1

Choose a value on the interval $x < - 7$ and see if this value makes the original inequality true.

$x = - 10$

Step 7.1.2

Replace $x$ with $- 10$ in the original inequality.

$\frac{6}{14 ((- 10) - 7)} > - \frac{8}{14 ((- 10) + 7)}$

Step 7.1.3

The left side $- 0.02521008$ is not greater than the right side $0.\overline{190476}$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $- 7 < x < 1$ to see if it makes the inequality true.

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Step 7.2.1

Choose a value on the interval $- 7 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 7.2.2

Replace $x$ with $0$ in the original inequality.

$\frac{6}{14 ((0) - 7)} > - \frac{8}{14 ((0) + 7)}$

Step 7.2.3

The left side $- 0.06122448$ is greater than the right side $- 0.08163265$ , which means that the given statement is always true.

True

Step 7.3

Test a value on the interval $1 < x < 7$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $1 < x < 7$ and see if this value makes the original inequality true.

$x = 4$

Step 7.3.2

Replace $x$ with $4$ in the original inequality.

$\frac{6}{14 ((4) - 7)} > - \frac{8}{14 ((4) + 7)}$

Step 7.3.3

The left side $- 0.\overline{142857}$ is not greater than the right side $- 0.\overline{051948}$ , which means that the given statement is false.

False

Step 7.4

Test a value on the interval $x > 7$ to see if it makes the inequality true.

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Step 7.4.1

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Step 7.4.2

Replace $x$ with $10$ in the original inequality.

$\frac{6}{14 ((10) - 7)} > - \frac{8}{14 ((10) + 7)}$

Step 7.4.3

The left side $0.\overline{142857}$ is greater than the right side $- 0.03361344$ , which means that the given statement is always true.

True

Step 7.5

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 7$ False

$- 7 < x < 1$ True

$1 < x < 7$ False

$x > 7$ True

$x < - 7$ False

$- 7 < x < 1$ True

$1 < x < 7$ False

$x > 7$ True

Step 8

The solution consists of all of the true intervals.

$- 7 < x < 1$ or $x > 7$

Step 9