Precalculus Examples

\frac{2 x + 1}{x + 4} + \frac{1 - x}{x - 3}

$\frac{2 x + 1}{x + 4} + \frac{1 - x}{x - 3}$

Step 1

To write

\frac{2 x + 1}{x + 4}

$\frac{2 x + 1}{x + 4}$ as a fraction with a common denominator, multiply by

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

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

\frac{2 x + 1}{x + 4} \cdot \frac{x - 3}{x - 3} + \frac{1 - x}{x - 3}

$\frac{2 x + 1}{x + 4} \cdot \frac{x - 3}{x - 3} + \frac{1 - x}{x - 3}$

Step 2

To write

\frac{1 - x}{x - 3}

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

\frac{x + 4}{x + 4}

$\frac{x + 4}{x + 4}$ .

\frac{2 x + 1}{x + 4} \cdot \frac{x - 3}{x - 3} + \frac{1 - x}{x - 3} \cdot \frac{x + 4}{x + 4}

$\frac{2 x + 1}{x + 4} \cdot \frac{x - 3}{x - 3} + \frac{1 - x}{x - 3} \cdot \frac{x + 4}{x + 4}$

Step 3

Write each expression with a common denominator of

(x + 4) (x - 3)

$(x + 4) (x - 3)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{2 x + 1}{x + 4}

$\frac{2 x + 1}{x + 4}$ by

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

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

\frac{(2 x + 1) (x - 3)}{(x + 4) (x - 3)} + \frac{1 - x}{x - 3} \cdot \frac{x + 4}{x + 4}

$\frac{(2 x + 1) (x - 3)}{(x + 4) (x - 3)} + \frac{1 - x}{x - 3} \cdot \frac{x + 4}{x + 4}$

Step 3.2

Multiply

\frac{1 - x}{x - 3}

$\frac{1 - x}{x - 3}$ by

\frac{x + 4}{x + 4}

$\frac{x + 4}{x + 4}$ .

\frac{(2 x + 1) (x - 3)}{(x + 4) (x - 3)} + \frac{(1 - x) (x + 4)}{(x - 3) (x + 4)}

$\frac{(2 x + 1) (x - 3)}{(x + 4) (x - 3)} + \frac{(1 - x) (x + 4)}{(x - 3) (x + 4)}$

Step 3.3

Reorder the factors of

(x - 3) (x + 4)

$(x - 3) (x + 4)$ .

\frac{(2 x + 1) (x - 3)}{(x + 4) (x - 3)} + \frac{(1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{(2 x + 1) (x - 3)}{(x + 4) (x - 3)} + \frac{(1 - x) (x + 4)}{(x + 4) (x - 3)}$

\frac{(2 x + 1) (x - 3)}{(x + 4) (x - 3)} + \frac{(1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{(2 x + 1) (x - 3)}{(x + 4) (x - 3)} + \frac{(1 - x) (x + 4)}{(x + 4) (x - 3)}$

Step 4

Combine the numerators over the common denominator.

\frac{(2 x + 1) (x - 3) + (1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{(2 x + 1) (x - 3) + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand

(2 x + 1) (x - 3)

$(2 x + 1) (x - 3)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

\frac{2 x (x - 3) + 1 (x - 3) + (1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{2 x (x - 3) + 1 (x - 3) + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

Step 5.1.2

Apply the distributive property.

\frac{2 x \cdot x + 2 x \cdot - 3 + 1 (x - 3) + (1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{2 x \cdot x + 2 x \cdot - 3 + 1 (x - 3) + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

Step 5.1.3

Apply the distributive property.

\frac{2 x \cdot x + 2 x \cdot - 3 + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{2 x \cdot x + 2 x \cdot - 3 + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

\frac{2 x \cdot x + 2 x \cdot - 3 + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{2 x \cdot x + 2 x \cdot - 3 + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

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

$x$ by

x

$x$ by adding the exponents.

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Step 5.2.1.1.1

Move

x

$x$ .

\frac{2 (x \cdot x) + 2 x \cdot - 3 + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{2 (x \cdot x) + 2 x \cdot - 3 + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

Step 5.2.1.1.2

Multiply

x

$x$ by

x

$x$ .

\frac{2 x^{2} + 2 x \cdot - 3 + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{2 x^{2} + 2 x \cdot - 3 + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

\frac{2 x^{2} + 2 x \cdot - 3 + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{2 x^{2} + 2 x \cdot - 3 + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

Step 5.2.1.2

Multiply

- 3

$- 3$ by

2

$2$ .

\frac{2 x^{2} - 6 x + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{2 x^{2} - 6 x + 1 x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

Step 5.2.1.3

Multiply

x

$x$ by

1

$1$ .

\frac{2 x^{2} - 6 x + x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{2 x^{2} - 6 x + x + 1 \cdot - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

Step 5.2.1.4

Multiply

- 3

$- 3$ by

1

$1$ .

\frac{2 x^{2} - 6 x + x - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}

$\frac{2 x^{2} - 6 x + x - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

Step 5.2.2

Add

$- 6 x$ and

$x$ .

$\frac{2 x^{2} - 5 x - 3 + (1 - x) (x + 4)}{(x + 4) (x - 3)}$

Step 5.3

Expand

$(1 - x) (x + 4)$ using the FOIL Method.

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Step 5.3.1

Apply the distributive property.

$\frac{2 x^{2} - 5 x - 3 + 1 (x + 4) - x (x + 4)}{(x + 4) (x - 3)}$

Step 5.3.2

Apply the distributive property.

$\frac{2 x^{2} - 5 x - 3 + 1 x + 1 \cdot 4 - x (x + 4)}{(x + 4) (x - 3)}$

Step 5.3.3

Apply the distributive property.

$\frac{2 x^{2} - 5 x - 3 + 1 x + 1 \cdot 4 - x \cdot x - x \cdot 4}{(x + 4) (x - 3)}$

Step 5.4

Simplify and combine like terms.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Multiply

$x$ by

$1$ .

$\frac{2 x^{2} - 5 x - 3 + x + 1 \cdot 4 - x \cdot x - x \cdot 4}{(x + 4) (x - 3)}$

Step 5.4.1.2

Multiply

$4$ by

$1$ .

$\frac{2 x^{2} - 5 x - 3 + x + 4 - x \cdot x - x \cdot 4}{(x + 4) (x - 3)}$

Step 5.4.1.3

Multiply

$x$ by

$x$ by adding the exponents.

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Step 5.4.1.3.1

Move

$x$ .

$\frac{2 x^{2} - 5 x - 3 + x + 4 - (x \cdot x) - x \cdot 4}{(x + 4) (x - 3)}$

Step 5.4.1.3.2

Multiply

$x$ by

$x$ .

$\frac{2 x^{2} - 5 x - 3 + x + 4 - x^{2} - x \cdot 4}{(x + 4) (x - 3)}$

Step 5.4.1.4

Multiply

$4$ by

$- 1$ .

$\frac{2 x^{2} - 5 x - 3 + x + 4 - x^{2} - 4 x}{(x + 4) (x - 3)}$

Step 5.4.2

Subtract

$4 x$ from

$x$ .

$\frac{2 x^{2} - 5 x - 3 - 3 x + 4 - x^{2}}{(x + 4) (x - 3)}$

Step 5.5

Subtract

$x^{2}$ from

$2 x^{2}$ .

$\frac{x^{2} - 5 x - 3 - 3 x + 4}{(x + 4) (x - 3)}$

Step 5.6

Subtract

$3 x$ from

$- 5 x$ .

$\frac{x^{2} - 8 x - 3 + 4}{(x + 4) (x - 3)}$

Step 5.7

Add

$- 3$ and

$4$ .

$\frac{x^{2} - 8 x + 1}{(x + 4) (x - 3)}$