Pre-Algebra Examples

$\frac{9}{x^{2} - 13 x + 42} - \frac{8}{x - 7} = \frac{1}{6 x - 36}$

Step 1

Simplify $\frac{9}{x^{2} - 13 x + 42} - \frac{8}{x - 7}$ .

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Step 1.1

Factor $x^{2} - 13 x + 42$ using the AC method.

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Step 1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $42$ and whose sum is $- 13$ .

$- 7, - 6$

Step 1.1.2

Write the factored form using these integers.

$\frac{9}{(x - 7) (x - 6)} - \frac{8}{x - 7} = \frac{1}{6 x - 36}$

Step 1.2

To write $- \frac{8}{x - 7}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{9}{(x - 7) (x - 6)} - \frac{8}{x - 7} \cdot \frac{x - 6}{x - 6} = \frac{1}{6 x - 36}$

Step 1.3

Simplify terms.

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Step 1.3.1

Multiply $\frac{8}{x - 7}$ by $\frac{x - 6}{x - 6}$ .

$\frac{9}{(x - 7) (x - 6)} - \frac{8 (x - 6)}{(x - 7) (x - 6)} = \frac{1}{6 x - 36}$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{9 - 8 (x - 6)}{(x - 7) (x - 6)} = \frac{1}{6 x - 36}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$\frac{9 - 8 x - 8 \cdot - 6}{(x - 7) (x - 6)} = \frac{1}{6 x - 36}$

Step 1.4.2

Multiply $- 8$ by $- 6$ .

$\frac{9 - 8 x + 48}{(x - 7) (x - 6)} = \frac{1}{6 x - 36}$

Step 1.4.3

Add $9$ and $48$ .

$\frac{- 8 x + 57}{(x - 7) (x - 6)} = \frac{1}{6 x - 36}$

Step 1.5

Simplify with factoring out.

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Step 1.5.1

Factor $- 1$ out of $- 8 x$ .

$\frac{- (8 x) + 57}{(x - 7) (x - 6)} = \frac{1}{6 x - 36}$

Step 1.5.2

Rewrite $57$ as $- 1 (- 57)$ .

$\frac{- (8 x) - 1 (- 57)}{(x - 7) (x - 6)} = \frac{1}{6 x - 36}$

Step 1.5.3

Factor $- 1$ out of $- (8 x) - 1 (- 57)$ .

$\frac{- (8 x - 57)}{(x - 7) (x - 6)} = \frac{1}{6 x - 36}$

Step 1.5.4

Simplify the expression.

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Step 1.5.4.1

Rewrite $- (8 x - 57)$ as $- 1 (8 x - 57)$ .

$\frac{- 1 (8 x - 57)}{(x - 7) (x - 6)} = \frac{1}{6 x - 36}$

Step 1.5.4.2

Move the negative in front of the fraction.

$- \frac{8 x - 57}{(x - 7) (x - 6)} = \frac{1}{6 x - 36}$

Step 2

Factor $6$ out of $6 x - 36$ .

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Step 2.1

Factor $6$ out of $6 x$ .

$- \frac{8 x - 57}{(x - 7) (x - 6)} = \frac{1}{6 (x) - 36}$

Step 2.2

Factor $6$ out of $- 36$ .

$- \frac{8 x - 57}{(x - 7) (x - 6)} = \frac{1}{6 x + 6 \cdot - 6}$

Step 2.3

Factor $6$ out of $6 x + 6 \cdot - 6$ .

$- \frac{8 x - 57}{(x - 7) (x - 6)} = \frac{1}{6 (x - 6)}$

Step 3

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

$x \approx 7.12244897$

Step 4