Pre-Algebra Examples

\frac{x}{x - 3} + \frac{5 x + 1}{x + 8}

$\frac{x}{x - 3} + \frac{5 x + 1}{x + 8}$

Step 1

To write

\frac{x}{x - 3}

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

\frac{x + 8}{x + 8}

$\frac{x + 8}{x + 8}$ .

\frac{x}{x - 3} \cdot \frac{x + 8}{x + 8} + \frac{5 x + 1}{x + 8}

$\frac{x}{x - 3} \cdot \frac{x + 8}{x + 8} + \frac{5 x + 1}{x + 8}$

Step 2

To write

\frac{5 x + 1}{x + 8}

$\frac{5 x + 1}{x + 8}$ as a fraction with a common denominator, multiply by

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

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

\frac{x}{x - 3} \cdot \frac{x + 8}{x + 8} + \frac{5 x + 1}{x + 8} \cdot \frac{x - 3}{x - 3}

$\frac{x}{x - 3} \cdot \frac{x + 8}{x + 8} + \frac{5 x + 1}{x + 8} \cdot \frac{x - 3}{x - 3}$

Step 3

Write each expression with a common denominator of

(x - 3) (x + 8)

$(x - 3) (x + 8)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{x}{x - 3}

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

\frac{x + 8}{x + 8}

$\frac{x + 8}{x + 8}$ .

\frac{x (x + 8)}{(x - 3) (x + 8)} + \frac{5 x + 1}{x + 8} \cdot \frac{x - 3}{x - 3}

$\frac{x (x + 8)}{(x - 3) (x + 8)} + \frac{5 x + 1}{x + 8} \cdot \frac{x - 3}{x - 3}$

Step 3.2

Multiply

\frac{5 x + 1}{x + 8}

$\frac{5 x + 1}{x + 8}$ by

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

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

\frac{x (x + 8)}{(x - 3) (x + 8)} + \frac{(5 x + 1) (x - 3)}{(x + 8) (x - 3)}

$\frac{x (x + 8)}{(x - 3) (x + 8)} + \frac{(5 x + 1) (x - 3)}{(x + 8) (x - 3)}$

Step 3.3

Reorder the factors of

(x - 3) (x + 8)

$(x - 3) (x + 8)$ .

\frac{x (x + 8)}{(x + 8) (x - 3)} + \frac{(5 x + 1) (x - 3)}{(x + 8) (x - 3)}

$\frac{x (x + 8)}{(x + 8) (x - 3)} + \frac{(5 x + 1) (x - 3)}{(x + 8) (x - 3)}$

\frac{x (x + 8)}{(x + 8) (x - 3)} + \frac{(5 x + 1) (x - 3)}{(x + 8) (x - 3)}

$\frac{x (x + 8)}{(x + 8) (x - 3)} + \frac{(5 x + 1) (x - 3)}{(x + 8) (x - 3)}$

Step 4

Combine the numerators over the common denominator.

\frac{x (x + 8) + (5 x + 1) (x - 3)}{(x + 8) (x - 3)}

$\frac{x (x + 8) + (5 x + 1) (x - 3)}{(x + 8) (x - 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

\frac{x \cdot x + x \cdot 8 + (5 x + 1) (x - 3)}{(x + 8) (x - 3)}

$\frac{x \cdot x + x \cdot 8 + (5 x + 1) (x - 3)}{(x + 8) (x - 3)}$

Step 5.2

Multiply

x

$x$ by

x

$x$ .

\frac{x^{2} + x \cdot 8 + (5 x + 1) (x - 3)}{(x + 8) (x - 3)}

$\frac{x^{2} + x \cdot 8 + (5 x + 1) (x - 3)}{(x + 8) (x - 3)}$

Step 5.3

Move

8

$8$ to the left of

x

$x$ .

\frac{x^{2} + 8 \cdot x + (5 x + 1) (x - 3)}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 \cdot x + (5 x + 1) (x - 3)}{(x + 8) (x - 3)}$

Step 5.4

Expand

(5 x + 1) (x - 3)

$(5 x + 1) (x - 3)$ using the FOIL Method.

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Step 5.4.1

Apply the distributive property.

\frac{x^{2} + 8 x + 5 x (x - 3) + 1 (x - 3)}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x (x - 3) + 1 (x - 3)}{(x + 8) (x - 3)}$

Step 5.4.2

Apply the distributive property.

\frac{x^{2} + 8 x + 5 x \cdot x + 5 x \cdot - 3 + 1 (x - 3)}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x \cdot x + 5 x \cdot - 3 + 1 (x - 3)}{(x + 8) (x - 3)}$

Step 5.4.3

Apply the distributive property.

\frac{x^{2} + 8 x + 5 x \cdot x + 5 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x \cdot x + 5 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}$

\frac{x^{2} + 8 x + 5 x \cdot x + 5 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x \cdot x + 5 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}$

Step 5.5

Simplify and combine like terms.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 5.5.1.1.1

Move

x

$x$ .

\frac{x^{2} + 8 x + 5 (x \cdot x) + 5 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 (x \cdot x) + 5 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}$

Step 5.5.1.1.2

Multiply

x

$x$ by

x

$x$ .

\frac{x^{2} + 8 x + 5 x^{2} + 5 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x^{2} + 5 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}$

\frac{x^{2} + 8 x + 5 x^{2} + 5 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x^{2} + 5 x \cdot - 3 + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}$

Step 5.5.1.2

Multiply

- 3

$- 3$ by

5

$5$ .

\frac{x^{2} + 8 x + 5 x^{2} - 15 x + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x^{2} - 15 x + 1 x + 1 \cdot - 3}{(x + 8) (x - 3)}$

Step 5.5.1.3

Multiply

x

$x$ by

1

$1$ .

\frac{x^{2} + 8 x + 5 x^{2} - 15 x + x + 1 \cdot - 3}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x^{2} - 15 x + x + 1 \cdot - 3}{(x + 8) (x - 3)}$

Step 5.5.1.4

Multiply

- 3

$- 3$ by

1

$1$ .

\frac{x^{2} + 8 x + 5 x^{2} - 15 x + x - 3}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x^{2} - 15 x + x - 3}{(x + 8) (x - 3)}$

\frac{x^{2} + 8 x + 5 x^{2} - 15 x + x - 3}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x^{2} - 15 x + x - 3}{(x + 8) (x - 3)}$

Step 5.5.2

Add

- 15 x

$- 15 x$ and

x

$x$ .

\frac{x^{2} + 8 x + 5 x^{2} - 14 x - 3}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x^{2} - 14 x - 3}{(x + 8) (x - 3)}$

\frac{x^{2} + 8 x + 5 x^{2} - 14 x - 3}{(x + 8) (x - 3)}

$\frac{x^{2} + 8 x + 5 x^{2} - 14 x - 3}{(x + 8) (x - 3)}$

Step 5.6

Add

x^{2}

$x^{2}$ and

5 x^{2}

$5 x^{2}$ .

\frac{6 x^{2} + 8 x - 14 x - 3}{(x + 8) (x - 3)}

$\frac{6 x^{2} + 8 x - 14 x - 3}{(x + 8) (x - 3)}$

Step 5.7

Subtract

14 x

$14 x$ from

8 x

$8 x$ .

\frac{6 x^{2} - 6 x - 3}{(x + 8) (x - 3)}

$\frac{6 x^{2} - 6 x - 3}{(x + 8) (x - 3)}$

Step 5.8

Factor

3

$3$ out of

6 x^{2} - 6 x - 3

$6 x^{2} - 6 x - 3$ .

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Step 5.8.1

Factor

3

$3$ out of

6 x^{2}

$6 x^{2}$ .

\frac{3 (2 x^{2}) - 6 x - 3}{(x + 8) (x - 3)}

$\frac{3 (2 x^{2}) - 6 x - 3}{(x + 8) (x - 3)}$

Step 5.8.2

Factor

3

$3$ out of

- 6 x

$- 6 x$ .

\frac{3 (2 x^{2}) + 3 (- 2 x) - 3}{(x + 8) (x - 3)}

$\frac{3 (2 x^{2}) + 3 (- 2 x) - 3}{(x + 8) (x - 3)}$

Step 5.8.3

Factor

3

$3$ out of

- 3

$- 3$ .

\frac{3 (2 x^{2}) + 3 (- 2 x) + 3 (- 1)}{(x + 8) (x - 3)}

$\frac{3 (2 x^{2}) + 3 (- 2 x) + 3 (- 1)}{(x + 8) (x - 3)}$

Step 5.8.4

Factor

3

$3$ out of

3 (2 x^{2}) + 3 (- 2 x)

$3 (2 x^{2}) + 3 (- 2 x)$ .

\frac{3 (2 x^{2} - 2 x) + 3 (- 1)}{(x + 8) (x - 3)}

$\frac{3 (2 x^{2} - 2 x) + 3 (- 1)}{(x + 8) (x - 3)}$

Step 5.8.5

Factor

3

$3$ out of

3 (2 x^{2} - 2 x) + 3 (- 1)

$3 (2 x^{2} - 2 x) + 3 (- 1)$ .

\frac{3 (2 x^{2} - 2 x - 1)}{(x + 8) (x - 3)}

$\frac{3 (2 x^{2} - 2 x - 1)}{(x + 8) (x - 3)}$

\frac{3 (2 x^{2} - 2 x - 1)}{(x + 8) (x - 3)}

$\frac{3 (2 x^{2} - 2 x - 1)}{(x + 8) (x - 3)}$

\frac{3 (2 x^{2} - 2 x - 1)}{(x + 8) (x - 3)}

$\frac{3 (2 x^{2} - 2 x - 1)}{(x + 8) (x - 3)}$