Algebra Examples

$\frac{x - 3}{x + 3} - \frac{1}{x - 3} + \frac{x + 2}{3 - x}$

Step 1

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

$\frac{x - 3}{x + 3} \cdot \frac{x - 3}{x - 3} - \frac{1}{x - 3} + \frac{x + 2}{3 - x}$

Step 2

To write $- \frac{1}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{x - 3}{x + 3} \cdot \frac{x - 3}{x - 3} - \frac{1}{x - 3} \cdot \frac{x + 3}{x + 3} + \frac{x + 2}{3 - x}$

Step 3

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

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Step 3.1

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

$\frac{(x - 3) (x - 3)}{(x + 3) (x - 3)} - \frac{1}{x - 3} \cdot \frac{x + 3}{x + 3} + \frac{x + 2}{3 - x}$

Step 3.2

Multiply $\frac{1}{x - 3}$ by $\frac{x + 3}{x + 3}$ .

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

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.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $x$ by $x$ .

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

Step 5.2.1.2

Move $- 3$ to the left of $x$ .

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

Step 5.2.1.3

Multiply $- 3$ by $- 3$ .

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

Step 5.2.2

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

$\frac{x^{2} - 6 x + 9 - (x + 3)}{(x + 3) (x - 3)} + \frac{x + 2}{3 - x}$

Step 5.3

Apply the distributive property.

$\frac{x^{2} - 6 x + 9 - x - 1 \cdot 3}{(x + 3) (x - 3)} + \frac{x + 2}{3 - x}$

Step 5.4

Multiply $- 1$ by $3$ .

$\frac{x^{2} - 6 x + 9 - x - 3}{(x + 3) (x - 3)} + \frac{x + 2}{3 - x}$

Step 5.5

Subtract $x$ from $- 6 x$ .

$\frac{x^{2} - 7 x + 9 - 3}{(x + 3) (x - 3)} + \frac{x + 2}{3 - x}$

Step 5.6

Subtract $3$ from $9$ .

$\frac{x^{2} - 7 x + 6}{(x + 3) (x - 3)} + \frac{x + 2}{3 - x}$

Step 5.7

Factor $x^{2} - 7 x + 6$ using the AC method.

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Step 5.7.1

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 $6$ and whose sum is $- 7$ .

$- 6, - 1$

Step 5.7.2

Write the factored form using these integers.

$\frac{(x - 6) (x - 1)}{(x + 3) (x - 3)} + \frac{x + 2}{3 - x}$

Step 6

Simplify with factoring out.

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Step 6.1

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{(x - 6) (x - 1)}{(x + 3) (x - 3)} + \frac{x + 2}{- 1 (- 3) - x}$

Step 6.2

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

$\frac{(x - 6) (x - 1)}{(x + 3) (x - 3)} + \frac{x + 2}{- 1 (- 3) - (x)}$

Step 6.3

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

$\frac{(x - 6) (x - 1)}{(x + 3) (x - 3)} + \frac{x + 2}{- 1 (- 3 + x)}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Move a negative from the denominator of $\frac{x + 2}{- 1 (- 3 + x)}$ to the numerator.

$\frac{(x - 6) (x - 1)}{(x + 3) (x - 3)} + \frac{- (x + 2)}{- 3 + x}$

Step 6.4.2

Reorder terms.

$\frac{(x - 6) (x - 1)}{(x + 3) (x - 3)} + \frac{- (x + 2)}{x - 3}$

Step 7

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

$\frac{(x - 6) (x - 1)}{(x + 3) (x - 3)} + \frac{- (x + 2)}{x - 3} \cdot \frac{x + 3}{x + 3}$

Step 8

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

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Step 8.1

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

$\frac{(x - 6) (x - 1)}{(x + 3) (x - 3)} + \frac{- (x + 2) (x + 3)}{(x - 3) (x + 3)}$

Step 8.2

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

$\frac{(x - 6) (x - 1)}{(x + 3) (x - 3)} + \frac{- (x + 2) (x + 3)}{(x + 3) (x - 3)}$

Step 9

Combine the numerators over the common denominator.

$\frac{(x - 6) (x - 1) - (x + 2) (x + 3)}{(x + 3) (x - 3)}$

Step 10

Simplify the numerator.

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Step 10.1

Expand $(x - 6) (x - 1)$ using the FOIL Method.

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Step 10.1.1

Apply the distributive property.

$\frac{x (x - 1) - 6 (x - 1) - (x + 2) (x + 3)}{(x + 3) (x - 3)}$

Step 10.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 1 - 6 (x - 1) - (x + 2) (x + 3)}{(x + 3) (x - 3)}$

Step 10.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 1 - 6 x - 6 \cdot - 1 - (x + 2) (x + 3)}{(x + 3) (x - 3)}$

Step 10.2

Simplify and combine like terms.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 1 - 6 x - 6 \cdot - 1 - (x + 2) (x + 3)}{(x + 3) (x - 3)}$

Step 10.2.1.2

Move $- 1$ to the left of $x$ .

$\frac{x^{2} - 1 \cdot x - 6 x - 6 \cdot - 1 - (x + 2) (x + 3)}{(x + 3) (x - 3)}$

Step 10.2.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{x^{2} - x - 6 x - 6 \cdot - 1 - (x + 2) (x + 3)}{(x + 3) (x - 3)}$

Step 10.2.1.4

Multiply $- 6$ by $- 1$ .

$\frac{x^{2} - x - 6 x + 6 - (x + 2) (x + 3)}{(x + 3) (x - 3)}$

Step 10.2.2

Subtract $6 x$ from $- x$ .

$\frac{x^{2} - 7 x + 6 - (x + 2) (x + 3)}{(x + 3) (x - 3)}$

Step 10.3

Apply the distributive property.

$\frac{x^{2} - 7 x + 6 + (- x - 1 \cdot 2) (x + 3)}{(x + 3) (x - 3)}$

Step 10.4

Multiply $- 1$ by $2$ .

$\frac{x^{2} - 7 x + 6 + (- x - 2) (x + 3)}{(x + 3) (x - 3)}$

Step 10.5

Expand $(- x - 2) (x + 3)$ using the FOIL Method.

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Step 10.5.1

Apply the distributive property.

$\frac{x^{2} - 7 x + 6 - x (x + 3) - 2 (x + 3)}{(x + 3) (x - 3)}$

Step 10.5.2

Apply the distributive property.

$\frac{x^{2} - 7 x + 6 - x \cdot x - x \cdot 3 - 2 (x + 3)}{(x + 3) (x - 3)}$

Step 10.5.3

Apply the distributive property.

$\frac{x^{2} - 7 x + 6 - x \cdot x - x \cdot 3 - 2 x - 2 \cdot 3}{(x + 3) (x - 3)}$

Step 10.6

Simplify and combine like terms.

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Step 10.6.1

Simplify each term.

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Step 10.6.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 10.6.1.1.1

Move $x$ .

$\frac{x^{2} - 7 x + 6 - (x \cdot x) - x \cdot 3 - 2 x - 2 \cdot 3}{(x + 3) (x - 3)}$

Step 10.6.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 7 x + 6 - x^{2} - x \cdot 3 - 2 x - 2 \cdot 3}{(x + 3) (x - 3)}$

Step 10.6.1.2

Multiply $3$ by $- 1$ .

$\frac{x^{2} - 7 x + 6 - x^{2} - 3 x - 2 x - 2 \cdot 3}{(x + 3) (x - 3)}$

Step 10.6.1.3

Multiply $- 2$ by $3$ .

$\frac{x^{2} - 7 x + 6 - x^{2} - 3 x - 2 x - 6}{(x + 3) (x - 3)}$

Step 10.6.2

Subtract $2 x$ from $- 3 x$ .

$\frac{x^{2} - 7 x + 6 - x^{2} - 5 x - 6}{(x + 3) (x - 3)}$

Step 10.7

Subtract $x^{2}$ from $x^{2}$ .

$\frac{- 7 x + 6 + 0 - 5 x - 6}{(x + 3) (x - 3)}$

Step 10.8

Add $- 7 x$ and $0$ .

$\frac{- 7 x + 6 - 5 x - 6}{(x + 3) (x - 3)}$

Step 10.9

Subtract $5 x$ from $- 7 x$ .

$\frac{- 12 x + 6 - 6}{(x + 3) (x - 3)}$

Step 10.10

Subtract $6$ from $6$ .

$\frac{- 12 x + 0}{(x + 3) (x - 3)}$

Step 10.11

Add $- 12 x$ and $0$ .

$\frac{- 12 x}{(x + 3) (x - 3)}$

Step 11

Move the negative in front of the fraction.

$- \frac{12 x}{(x + 3) (x - 3)}$