Pre-Algebra Examples

$\frac{x + 3}{x + 2} + \frac{x - 2}{x - 3}$

Step 1

To write $\frac{x + 3}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x + 3}{x + 2} \cdot \frac{x - 3}{x - 3} + \frac{x - 2}{x - 3}$

Step 2

To write $\frac{x - 2}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{x + 3}{x + 2} \cdot \frac{x - 3}{x - 3} + \frac{x - 2}{x - 3} \cdot \frac{x + 2}{x + 2}$

Step 3

Write each expression with a common denominator of $(x + 2) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x + 3}{x + 2}$ by $\frac{x - 3}{x - 3}$ .

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

Step 3.2

Multiply $\frac{x - 2}{x - 3}$ by $\frac{x + 2}{x + 2}$ .

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

Step 3.3

Reorder the factors of $(x - 3) (x + 2)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Expand $(x + 3) (x - 3)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Combine the opposite terms in $x \cdot x + x \cdot - 3 + 3 x + 3 \cdot - 3$ .

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Step 5.2.1

Reorder the factors in the terms $x \cdot - 3$ and $3 x$ .

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

Step 5.2.2

Add $- 3 x$ and $3 x$ .

$\frac{x \cdot x + 0 + 3 \cdot - 3 + (x - 2) (x + 2)}{(x + 2) (x - 3)}$

Step 5.2.3

Add $x \cdot x$ and $0$ .

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

Step 5.3

Simplify each term.

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Step 5.3.1

Multiply $x$ by $x$ .

$\frac{x^{2} + 3 \cdot - 3 + (x - 2) (x + 2)}{(x + 2) (x - 3)}$

Step 5.3.2

Multiply $3$ by $- 3$ .

$\frac{x^{2} - 9 + (x - 2) (x + 2)}{(x + 2) (x - 3)}$

Step 5.4

Expand $(x - 2) (x + 2)$ using the FOIL Method.

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Step 5.4.1

Apply the distributive property.

$\frac{x^{2} - 9 + x (x + 2) - 2 (x + 2)}{(x + 2) (x - 3)}$

Step 5.4.2

Apply the distributive property.

$\frac{x^{2} - 9 + x \cdot x + x \cdot 2 - 2 (x + 2)}{(x + 2) (x - 3)}$

Step 5.4.3

Apply the distributive property.

$\frac{x^{2} - 9 + x \cdot x + x \cdot 2 - 2 x - 2 \cdot 2}{(x + 2) (x - 3)}$

Step 5.5

Combine the opposite terms in $x \cdot x + x \cdot 2 - 2 x - 2 \cdot 2$ .

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Step 5.5.1

Reorder the factors in the terms $x \cdot 2$ and $- 2 x$ .

$\frac{x^{2} - 9 + x \cdot x + 2 x - 2 x - 2 \cdot 2}{(x + 2) (x - 3)}$

Step 5.5.2

Subtract $2 x$ from $2 x$ .

$\frac{x^{2} - 9 + x \cdot x + 0 - 2 \cdot 2}{(x + 2) (x - 3)}$

Step 5.5.3

Add $x \cdot x$ and $0$ .

$\frac{x^{2} - 9 + x \cdot x - 2 \cdot 2}{(x + 2) (x - 3)}$

Step 5.6

Simplify each term.

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Step 5.6.1

Multiply $x$ by $x$ .

$\frac{x^{2} - 9 + x^{2} - 2 \cdot 2}{(x + 2) (x - 3)}$

Step 5.6.2

Multiply $- 2$ by $2$ .

$\frac{x^{2} - 9 + x^{2} - 4}{(x + 2) (x - 3)}$

Step 5.7

Add $x^{2}$ and $x^{2}$ .

$\frac{2 x^{2} - 9 - 4}{(x + 2) (x - 3)}$

Step 5.8

Subtract $4$ from $- 9$ .

$\frac{2 x^{2} - 13}{(x + 2) (x - 3)}$