Algebra Examples

\frac{1}{x + h} - \frac{1}{x}

$\frac{1}{x + h} - \frac{1}{x}$

Step 1

To write

\frac{1}{x + h}

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

\frac{x}{x}

$\frac{x}{x}$ .

\frac{1}{x + h} \cdot \frac{x}{x} - \frac{1}{x}

$\frac{1}{x + h} \cdot \frac{x}{x} - \frac{1}{x}$

Step 2

To write

- \frac{1}{x}

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

\frac{x + h}{x + h}

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

\frac{1}{x + h} \cdot \frac{x}{x} - \frac{1}{x} \cdot \frac{x + h}{x + h}

$\frac{1}{x + h} \cdot \frac{x}{x} - \frac{1}{x} \cdot \frac{x + h}{x + h}$

Step 3

Write each expression with a common denominator of

(x + h) x

$(x + h) x$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{1}{x + h}

$\frac{1}{x + h}$ by

\frac{x}{x}

$\frac{x}{x}$ .

\frac{x}{(x + h) x} - \frac{1}{x} \cdot \frac{x + h}{x + h}

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

Step 3.2

Multiply

\frac{1}{x}

$\frac{1}{x}$ by

\frac{x + h}{x + h}

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

\frac{x}{(x + h) x} - \frac{x + h}{x (x + h)}

$\frac{x}{(x + h) x} - \frac{x + h}{x (x + h)}$

Step 3.3

Reorder the factors of

(x + h) x

$(x + h) x$ .

\frac{x}{x (x + h)} - \frac{x + h}{x (x + h)}

$\frac{x}{x (x + h)} - \frac{x + h}{x (x + h)}$

\frac{x}{x (x + h)} - \frac{x + h}{x (x + h)}

$\frac{x}{x (x + h)} - \frac{x + h}{x (x + h)}$

Step 4

Combine the numerators over the common denominator.

\frac{x - (x + h)}{x (x + h)}

$\frac{x - (x + h)}{x (x + h)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

\frac{x - x - h}{x (x + h)}

$\frac{x - x - h}{x (x + h)}$

Step 5.2

Subtract

x

$x$ from

x

$x$ .

\frac{0 - h}{x (x + h)}

$\frac{0 - h}{x (x + h)}$

Step 5.3

Subtract

h

$h$ from

0

$0$ .

\frac{- h}{x (x + h)}

$\frac{- h}{x (x + h)}$

\frac{- h}{x (x + h)}

$\frac{- h}{x (x + h)}$

Step 6

Move the negative in front of the fraction.

- \frac{h}{x (x + h)}

$- \frac{h}{x (x + h)}$