Pre-Algebra Examples

$\frac{8}{h - 12} + \frac{h}{h + 3} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1

Simplify $\frac{8}{h - 12} + \frac{h}{h + 3}$ .

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Step 1.1

To write $\frac{8}{h - 12}$ as a fraction with a common denominator, multiply by $\frac{h + 3}{h + 3}$ .

$\frac{8}{h - 12} \cdot \frac{h + 3}{h + 3} + \frac{h}{h + 3} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.2

To write $\frac{h}{h + 3}$ as a fraction with a common denominator, multiply by $\frac{h - 12}{h - 12}$ .

$\frac{8}{h - 12} \cdot \frac{h + 3}{h + 3} + \frac{h}{h + 3} \cdot \frac{h - 12}{h - 12} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.3

Write each expression with a common denominator of $(h - 12) (h + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{8}{h - 12}$ by $\frac{h + 3}{h + 3}$ .

$\frac{8 (h + 3)}{(h - 12) (h + 3)} + \frac{h}{h + 3} \cdot \frac{h - 12}{h - 12} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.3.2

Multiply $\frac{h}{h + 3}$ by $\frac{h - 12}{h - 12}$ .

$\frac{8 (h + 3)}{(h - 12) (h + 3)} + \frac{h (h - 12)}{(h + 3) (h - 12)} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.3.3

Reorder the factors of $(h - 12) (h + 3)$ .

$\frac{8 (h + 3)}{(h + 3) (h - 12)} + \frac{h (h - 12)}{(h + 3) (h - 12)} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{8 (h + 3) + h (h - 12)}{(h + 3) (h - 12)} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{8 h + 8 \cdot 3 + h (h - 12)}{(h + 3) (h - 12)} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.5.2

Multiply $8$ by $3$ .

$\frac{8 h + 24 + h (h - 12)}{(h + 3) (h - 12)} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.5.3

Apply the distributive property.

$\frac{8 h + 24 + h \cdot h + h \cdot - 12}{(h + 3) (h - 12)} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.5.4

Multiply $h$ by $h$ .

$\frac{8 h + 24 + h^{2} + h \cdot - 12}{(h + 3) (h - 12)} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.5.5

Move $- 12$ to the left of $h$ .

$\frac{8 h + 24 + h^{2} - 12 \cdot h}{(h + 3) (h - 12)} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.5.6

Subtract $12 h$ from $8 h$ .

$\frac{- 4 h + 24 + h^{2}}{(h + 3) (h - 12)} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 1.5.7

Reorder terms.

$\frac{h^{2} - 4 h + 24}{(h + 3) (h - 12)} = \frac{5 h + 4}{h^{2} - 9 h - 36}$

Step 2

Factor $h^{2} - 9 h - 36$ using the AC method.

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Step 2.1

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 $- 36$ and whose sum is $- 9$ .

$- 12, 3$

Step 2.2

Write the factored form using these integers.

$\frac{h^{2} - 4 h + 24}{(h + 3) (h - 12)} = \frac{5 h + 4}{(h - 12) (h + 3)}$

Step 3

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

$h = 4, 5$

Step 4