Pre-Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor $x$ out of $x^{2} + 5 x$ .

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Step 1.1.1

Factor $x$ out of $x^{2}$ .

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

Step 1.1.2

Factor $x$ out of $5 x$ .

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

Step 1.1.3

Factor $x$ out of $x \cdot x + x \cdot 5$ .

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

Step 1.2

Factor $x^{2} + 4 x - 5$ using the AC method.

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Step 1.2.1

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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 1.2.2

Write the factored form using these integers.

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

Step 1.3

Cancel the common factor of $x + 5$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x - 1) x$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

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

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

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

Step 6.2.1.2

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

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

Step 6.2.1.3

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

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

Step 6.2.1.4

Multiply $x$ by $1$ .

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

Step 6.2.1.5

Multiply $- 1$ by $1$ .

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

Step 6.2.2

Add $- x$ and $x$ .

$\frac{x^{2} + 0 - 1 + x \cdot x}{x (x - 1)}$

Step 6.2.3

Add $x^{2}$ and $0$ .

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

Step 6.3

Multiply $x$ by $x$ .

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

Step 6.4

Add $x^{2}$ and $x^{2}$ .

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