Precalculus Examples

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

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

Step 1

To write

\frac{2}{x - 2}

$\frac{2}{x - 2}$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

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

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

Step 2

To write

- \frac{2}{3}

$- \frac{2}{3}$ as a fraction with a common denominator, multiply by

\frac{x - 2}{x - 2}

$\frac{x - 2}{x - 2}$ .

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

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

Step 3

Write each expression with a common denominator of

(x - 2) \cdot 3

$(x - 2) \cdot 3$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{2}{x - 2}

$\frac{2}{x - 2}$ by

\frac{3}{3}

$\frac{3}{3}$ .

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

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

Step 3.2

Multiply

\frac{2}{3}

$\frac{2}{3}$ by

\frac{x - 2}{x - 2}

$\frac{x - 2}{x - 2}$ .

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

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

Step 3.3

Reorder the factors of

(x - 2) \cdot 3

$(x - 2) \cdot 3$ .

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

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

Step 5

Simplify the numerator.

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Step 5.1

Factor

2

$2$ out of

2 \cdot 3 - 2 (x - 2)

$2 \cdot 3 - 2 (x - 2)$ .

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Step 5.1.1

Factor

2

$2$ out of

2 \cdot 3

$2 \cdot 3$ .

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

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

Step 5.1.2

Factor

2

$2$ out of

- 2 (x - 2)

$- 2 (x - 2)$ .

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

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

Step 5.1.3

Factor

2

$2$ out of

2 (3) + 2 (- (x - 2))

$2 (3) + 2 (- (x - 2))$ .

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

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

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

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

Step 5.2

Apply the distributive property.

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

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

Step 5.3

Multiply

- 1

$- 1$ by

- 2

$- 2$ .

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

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

Step 5.4

Add

3

$3$ and

2

$2$ .

\frac{2 (- x + 5)}{3 (x - 2)}

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

\frac{2 (- x + 5)}{3 (x - 2)}

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

Step 6

Simplify with factoring out.

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Step 6.1

Factor

- 1

$- 1$ out of

- x

$- x$ .

\frac{2 (- (x) + 5)}{3 (x - 2)}

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

Step 6.2

Rewrite

5

$5$ as

- 1 (- 5)

$- 1 (- 5)$ .

\frac{2 (- (x) - 1 (- 5))}{3 (x - 2)}

$\frac{2 (- (x) - 1 (- 5))}{3 (x - 2)}$

Step 6.3

Factor

- 1

$- 1$ out of

- (x) - 1 (- 5)

$- (x) - 1 (- 5)$ .

\frac{2 (- (x - 5))}{3 (x - 2)}

$\frac{2 (- (x - 5))}{3 (x - 2)}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite

- (x - 5)

$- (x - 5)$ as

- 1 (x - 5)

$- 1 (x - 5)$ .

\frac{2 (- 1 (x - 5))}{3 (x - 2)}

$\frac{2 (- 1 (x - 5))}{3 (x - 2)}$

Step 6.4.2

Move the negative in front of the fraction.

- \frac{2 (x - 5)}{3 (x - 2)}

$- \frac{2 (x - 5)}{3 (x - 2)}$

- \frac{2 (x - 5)}{3 (x - 2)}

$- \frac{2 (x - 5)}{3 (x - 2)}$

- \frac{2 (x - 5)}{3 (x - 2)}

$- \frac{2 (x - 5)}{3 (x - 2)}$

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