Algebra Examples

$x + 1 = \frac{4}{x + 1}$

Step 1

Subtract $1$ from both sides of the equation.

$x = \frac{4}{x + 1} - 1$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x + 1, 1$

Step 2.2

Remove parentheses.

$1, x + 1, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x + 1$

Step 3

Multiply each term in $x = \frac{4}{x + 1} - 1$ by $x + 1$ to eliminate the fractions.

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Step 3.1

Multiply each term in $x = \frac{4}{x + 1} - 1$ by $x + 1$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Multiply $x$ by $x$ .

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

Step 3.2.2.2

Multiply $x$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $x + 1$ .

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Step 3.3.1.1.1

Cancel the common factor.

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

Step 3.3.1.1.2

Rewrite the expression.

$x^{2} + x = 4 - (x + 1)$

Step 3.3.1.2

Apply the distributive property.

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

Step 3.3.1.3

Multiply $- 1$ by $1$ .

$x^{2} + x = 4 - x - 1$

Step 3.3.2

Subtract $1$ from $4$ .

$x^{2} + x = - x + 3$

Step 4

Solve the equation.

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Add $x$ to both sides of the equation.

$x^{2} + x + x = 3$

Step 4.1.2

Add $x$ and $x$ .

$x^{2} + 2 x = 3$

Step 4.2

Subtract $3$ from both sides of the equation.

$x^{2} + 2 x - 3 = 0$

Step 4.3

Factor $x^{2} + 2 x - 3$ using the AC method.

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Step 4.3.1

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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 4.3.2

Write the factored form using these integers.

$(x - 1) (x + 3) = 0$

Step 4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 1 = 0$

$x + 3 = 0$

Step 4.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.6

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 4.6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.7

The final solution is all the values that make $(x - 1) (x + 3) = 0$ true.

$x = 1, - 3$