Pre-Algebra Examples

$\frac{30 - 17 r}{r^{2} - 36} - \frac{r}{6 - r} = \frac{11}{r + 6}$

Step 1

Simplify $\frac{30 - 17 r}{r^{2} - 36} - \frac{r}{6 - r}$ .

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $36$ as $6^{2}$ .

$\frac{30 - 17 r}{r^{2} - 6^{2}} - \frac{r}{6 - r} = \frac{11}{r + 6}$

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r$ and $b = 6$ .

$\frac{30 - 17 r}{(r + 6) (r - 6)} - \frac{r}{6 - r} = \frac{11}{r + 6}$

Step 1.2

Simplify with factoring out.

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Step 1.2.1

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{30 - 17 r}{(r + 6) (r - 6)} - \frac{r}{- 1 (- 6) - r} = \frac{11}{r + 6}$

Step 1.2.2

Factor $- 1$ out of $- r$ .

$\frac{30 - 17 r}{(r + 6) (r - 6)} - \frac{r}{- 1 (- 6) - (r)} = \frac{11}{r + 6}$

Step 1.2.3

Factor $- 1$ out of $- 1 (- 6) - (r)$ .

$\frac{30 - 17 r}{(r + 6) (r - 6)} - \frac{r}{- 1 (- 6 + r)} = \frac{11}{r + 6}$

Step 1.2.4

Reorder terms.

$\frac{30 - 17 r}{(r + 6) (r - 6)} - \frac{r}{- 1 (r - 6)} = \frac{11}{r + 6}$

Step 1.3

To write $\frac{30 - 17 r}{(r + 6) (r - 6)}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$\frac{30 - 17 r}{(r + 6) (r - 6)} \cdot \frac{- 1}{- 1} - \frac{r}{- 1 (r - 6)} = \frac{11}{r + 6}$

Step 1.4

To write $- \frac{r}{- 1 (r - 6)}$ as a fraction with a common denominator, multiply by $\frac{r + 6}{r + 6}$ .

$\frac{30 - 17 r}{(r + 6) (r - 6)} \cdot \frac{- 1}{- 1} - \frac{r}{- 1 (r - 6)} \cdot \frac{r + 6}{r + 6} = \frac{11}{r + 6}$

Step 1.5

Write each expression with a common denominator of $(r + 6) (r - 6) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

Multiply $\frac{30 - 17 r}{(r + 6) (r - 6)}$ by $\frac{- 1}{- 1}$ .

$\frac{(30 - 17 r) \cdot - 1}{(r + 6) (r - 6) \cdot - 1} - \frac{r}{- 1 (r - 6)} \cdot \frac{r + 6}{r + 6} = \frac{11}{r + 6}$

Step 1.5.2

Multiply $\frac{r}{- 1 (r - 6)}$ by $\frac{r + 6}{r + 6}$ .

$\frac{(30 - 17 r) \cdot - 1}{(r + 6) (r - 6) \cdot - 1} - \frac{r (r + 6)}{- 1 (r - 6) (r + 6)} = \frac{11}{r + 6}$

Step 1.5.3

Reorder the factors of $(r + 6) (r - 6) \cdot - 1$ .

$\frac{(30 - 17 r) \cdot - 1}{- (r + 6) (r - 6)} - \frac{r (r + 6)}{- 1 (r - 6) (r + 6)} = \frac{11}{r + 6}$

Step 1.5.4

Reorder the factors of $- 1 (r - 6) (r + 6)$ .

$\frac{(30 - 17 r) \cdot - 1}{- (r + 6) (r - 6)} - \frac{r (r + 6)}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.6

Combine the numerators over the common denominator.

$\frac{(30 - 17 r) \cdot - 1 - r (r + 6)}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Apply the distributive property.

$\frac{30 \cdot - 1 - 17 r \cdot - 1 - r (r + 6)}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.2

Multiply $30$ by $- 1$ .

$\frac{- 30 - 17 r \cdot - 1 - r (r + 6)}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.3

Multiply $- 1$ by $- 17$ .

$\frac{- 30 + 17 r - r (r + 6)}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.4

Apply the distributive property.

$\frac{- 30 + 17 r - r \cdot r - r \cdot 6}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.5

Multiply $r$ by $r$ by adding the exponents.

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Step 1.7.5.1

Move $r$ .

$\frac{- 30 + 17 r - (r \cdot r) - r \cdot 6}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.5.2

Multiply $r$ by $r$ .

$\frac{- 30 + 17 r - r^{2} - r \cdot 6}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.6

Multiply $6$ by $- 1$ .

$\frac{- 30 + 17 r - r^{2} - 6 r}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.7

Subtract $6 r$ from $17 r$ .

$\frac{- 30 - r^{2} + 11 r}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.8

Reorder terms.

$\frac{- r^{2} + 11 r - 30}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.9

Factor by grouping.

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Step 1.7.9.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 30 = 30$ and whose sum is $b = 11$ .

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Step 1.7.9.1.1

Factor $11$ out of $11 r$ .

$\frac{- r^{2} + 11 (r) - 30}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.9.1.2

Rewrite $11$ as $5$ plus $6$

$\frac{- r^{2} + (5 + 6) r - 30}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.9.1.3

Apply the distributive property.

$\frac{- r^{2} + 5 r + 6 r - 30}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.9.2

Factor out the greatest common factor from each group.

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Step 1.7.9.2.1

Group the first two terms and the last two terms.

$\frac{(- r^{2} + 5 r) + 6 r - 30}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.9.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{r (- r + 5) - 6 (- r + 5)}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.7.9.3

Factor the polynomial by factoring out the greatest common factor, $- r + 5$ .

$\frac{(- r + 5) (r - 6)}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.8

Simplify terms.

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Step 1.8.1

Cancel the common factor of $r - 6$ .

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Step 1.8.1.1

Cancel the common factor.

$\frac{(- r + 5) (r - 6)}{- (r + 6) (r - 6)} = \frac{11}{r + 6}$

Step 1.8.1.2

Rewrite the expression.

$\frac{- r + 5}{- (r + 6)} = \frac{11}{r + 6}$

Step 1.8.2

Move the negative in front of the fraction.

$- \frac{- r + 5}{r + 6} = \frac{11}{r + 6}$

Step 1.8.3

Factor $- 1$ out of $- r$ .

$- \frac{- (r) + 5}{r + 6} = \frac{11}{r + 6}$

Step 1.8.4

Rewrite $5$ as $- 1 (- 5)$ .

$- \frac{- (r) - 1 (- 5)}{r + 6} = \frac{11}{r + 6}$

Step 1.8.5

Factor $- 1$ out of $- (r) - 1 (- 5)$ .

$- \frac{- (r - 5)}{r + 6} = \frac{11}{r + 6}$

Step 1.8.6

Simplify the expression.

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Step 1.8.6.1

Rewrite $- (r - 5)$ as $- 1 (r - 5)$ .

$- \frac{- 1 (r - 5)}{r + 6} = \frac{11}{r + 6}$

Step 1.8.6.2

Move the negative in front of the fraction.

$- - \frac{r - 5}{r + 6} = \frac{11}{r + 6}$

Step 1.8.6.3

Multiply $- 1$ by $- 1$ .

$1 \frac{r - 5}{r + 6} = \frac{11}{r + 6}$

Step 1.8.6.4

Multiply $\frac{r - 5}{r + 6}$ by $1$ .

$\frac{r - 5}{r + 6} = \frac{11}{r + 6}$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$r \approx 16$

Step 3