Algebra Examples

$\frac{x + 1}{x} - \frac{x}{x + 1} = 0$

Step 1

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

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Step 1.1

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

$\frac{x + 1}{x} \cdot \frac{x + 1}{x + 1} - \frac{x}{x + 1} = 0$

Step 1.2

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

$\frac{x + 1}{x} \cdot \frac{x + 1}{x + 1} - \frac{x}{x + 1} \cdot \frac{x}{x} = 0$

Step 1.3

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 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

Raise $x + 1$ to the power of $1$ .

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

Step 1.5.2

Raise $x + 1$ to the power of $1$ .

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

Step 1.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.5.4

Add $1$ and $1$ .

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

Step 1.5.5

Rewrite $x \cdot x$ as $x^{2}$ .

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

Step 1.5.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + 1$ and $b = x$ .

$\frac{(x + 1 + x) (x + 1 - x)}{x (x + 1)} = 0$

Step 1.5.7

Simplify.

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Step 1.5.7.1

Add $x$ and $x$ .

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

Step 1.5.7.2

Subtract $x$ from $x$ .

$\frac{(2 x + 1) (0 + 1)}{x (x + 1)} = 0$

Step 1.5.7.3

Add $0$ and $1$ .

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

Step 1.6

Multiply $2 x + 1$ by $1$ .

$\frac{2 x + 1}{x (x + 1)} = 0$

Step 2

Set the numerator equal to zero.

$2 x + 1 = 0$

Step 3

Solve the equation for $x$ .

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Step 3.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 3.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}$

Decimal Form:

$x = - 0.5$