Algebra Examples

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

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$x + 1, 1, 1$

Step 1.2

Remove parentheses.

$x + 1, 1, 1$

Step 1.3

The LCM of one and any expression is the expression.

$x + 1$

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x + 1$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

$4 x - 1 = x (x + 1) - (x + 1)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Apply the distributive property.

$4 x - 1 = x \cdot x + x \cdot 1 - (x + 1)$

Step 2.3.1.2

Multiply $x$ by $x$ .

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

Step 2.3.1.3

Multiply $x$ by $1$ .

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

Step 2.3.1.4

Apply the distributive property.

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

Step 2.3.1.5

Multiply $- 1$ by $1$ .

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

Step 2.3.2

Combine the opposite terms in $x^{2} + x - x - 1$ .

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Step 2.3.2.1

Subtract $x$ from $x$ .

$4 x - 1 = x^{2} + 0 - 1$

Step 2.3.2.2

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

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

Step 3

Solve the equation.

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Step 3.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 3.2

Add $1$ to both sides of the equation.

$4 x - 1 - x^{2} + 1 = 0$

Step 3.3

Combine the opposite terms in $4 x - 1 - x^{2} + 1$ .

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Step 3.3.1

Add $- 1$ and $1$ .

$4 x - x^{2} + 0 = 0$

Step 3.3.2

Add $4 x - x^{2}$ and $0$ .

$4 x - x^{2} = 0$

Step 3.4

Factor $x$ out of $4 x - x^{2}$ .

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Step 3.4.1

Factor $x$ out of $4 x$ .

$x \cdot 4 - x^{2} = 0$

Step 3.4.2

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

$x \cdot 4 + x (- x) = 0$

Step 3.4.3

Factor $x$ out of $x \cdot 4 + x (- x)$ .

$x (4 - x) = 0$

Step 3.5

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

$x = 0$

$4 - x = 0$

Step 3.6

Set $x$ equal to $0$ .

$x = 0$

Step 3.7

Set $4 - x$ equal to $0$ and solve for $x$ .

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Step 3.7.1

Set $4 - x$ equal to $0$ .

$4 - x = 0$

Step 3.7.2

Solve $4 - x = 0$ for $x$ .

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Step 3.7.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 3.7.2.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 3.7.2.2.1

Divide each term in $- x = - 4$ by $- 1$ .

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

Step 3.7.2.2.2

Simplify the left side.

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Step 3.7.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.7.2.2.2.2

Divide $x$ by $1$ .

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

Step 3.7.2.2.3

Simplify the right side.

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Step 3.7.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 3.8

The final solution is all the values that make $x (4 - x) = 0$ true.

$x = 0, 4$