Algebra Examples

$\frac{- 1}{(x + 3) (x - 4)} = \frac{1}{2 x + 1}$

Step 1

Subtract $\frac{1}{2 x + 1}$ from both sides of the equation.

$\frac{- 1}{(x + 3) (x - 4)} - \frac{1}{2 x + 1} = 0$

Step 2

Simplify $\frac{- 1}{(x + 3) (x - 4)} - \frac{1}{2 x + 1}$ .

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Step 2.1

Move the negative in front of the fraction.

$- \frac{1}{(x + 3) (x - 4)} - \frac{1}{2 x + 1} = 0$

Step 2.2

To write $- \frac{1}{(x + 3) (x - 4)}$ as a fraction with a common denominator, multiply by $\frac{2 x + 1}{2 x + 1}$ .

$- \frac{1}{(x + 3) (x - 4)} \cdot \frac{2 x + 1}{2 x + 1} - \frac{1}{2 x + 1} = 0$

Step 2.3

To write $- \frac{1}{2 x + 1}$ as a fraction with a common denominator, multiply by $\frac{(x + 3) (x - 4)}{(x + 3) (x - 4)}$ .

$- \frac{1}{(x + 3) (x - 4)} \cdot \frac{2 x + 1}{2 x + 1} - \frac{1}{2 x + 1} \cdot \frac{(x + 3) (x - 4)}{(x + 3) (x - 4)} = 0$

Step 2.4

Write each expression with a common denominator of $(x + 3) (x - 4) (2 x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{1}{(x + 3) (x - 4)}$ by $\frac{2 x + 1}{2 x + 1}$ .

$- \frac{2 x + 1}{(x + 3) (x - 4) (2 x + 1)} - \frac{1}{2 x + 1} \cdot \frac{(x + 3) (x - 4)}{(x + 3) (x - 4)} = 0$

Step 2.4.2

Multiply $\frac{1}{2 x + 1}$ by $\frac{(x + 3) (x - 4)}{(x + 3) (x - 4)}$ .

$- \frac{2 x + 1}{(x + 3) (x - 4) (2 x + 1)} - \frac{(x + 3) (x - 4)}{(2 x + 1) ((x + 3) (x - 4))} = 0$

Step 2.4.3

Reorder the factors of $(x + 3) (x - 4) (2 x + 1)$ .

$- \frac{2 x + 1}{(2 x + 1) (x + 3) (x - 4)} - \frac{(x + 3) (x - 4)}{(2 x + 1) ((x + 3) (x - 4))} = 0$

Step 2.4.4

Reorder the factors of $(2 x + 1) ((x + 3) (x - 4))$ .

$- \frac{2 x + 1}{(2 x + 1) (x + 3) (x - 4)} - \frac{(x + 3) (x - 4)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{- (2 x + 1) - (x + 3) (x - 4)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Factor $- 1$ out of $- (2 x + 1) - (x + 3) (x - 4)$ .

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Step 2.6.1.1

Reorder $- (2 x + 1)$ and $- (x + 3) (x - 4)$ .

$\frac{- (x + 3) (x - 4) - (2 x + 1)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6.1.2

Factor $- 1$ out of $- (x + 3) (x - 4)$ .

$\frac{- ((x + 3) (x - 4)) - (2 x + 1)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6.1.3

Factor $- 1$ out of $- ((x + 3) (x - 4)) - (2 x + 1)$ .

$\frac{- ((x + 3) (x - 4) + 2 x + 1)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6.2

Expand $(x + 3) (x - 4)$ using the FOIL Method.

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Step 2.6.2.1

Apply the distributive property.

$\frac{- (x (x - 4) + 3 (x - 4) + 2 x + 1)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6.2.2

Apply the distributive property.

$\frac{- (x \cdot x + x \cdot - 4 + 3 (x - 4) + 2 x + 1)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6.2.3

Apply the distributive property.

$\frac{- (x \cdot x + x \cdot - 4 + 3 x + 3 \cdot - 4 + 2 x + 1)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6.3

Simplify and combine like terms.

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Step 2.6.3.1

Simplify each term.

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Step 2.6.3.1.1

Multiply $x$ by $x$ .

$\frac{- (x^{2} + x \cdot - 4 + 3 x + 3 \cdot - 4 + 2 x + 1)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6.3.1.2

Move $- 4$ to the left of $x$ .

$\frac{- (x^{2} - 4 \cdot x + 3 x + 3 \cdot - 4 + 2 x + 1)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6.3.1.3

Multiply $3$ by $- 4$ .

$\frac{- (x^{2} - 4 x + 3 x - 12 + 2 x + 1)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6.3.2

Add $- 4 x$ and $3 x$ .

$\frac{- (x^{2} - x - 12 + 2 x + 1)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6.4

Add $- x$ and $2 x$ .

$\frac{- (x^{2} + x - 12 + 1)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.6.5

Add $- 12$ and $1$ .

$\frac{- (x^{2} + x - 11)}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 2.7

Move the negative in front of the fraction.

$- \frac{x^{2} + x - 11}{(2 x + 1) (x + 3) (x - 4)} = 0$

Step 3

Set the denominator in $\frac{x^{2} + x - 11}{(2 x + 1) (x + 3) (x - 4)}$ equal to $0$ to find where the expression is undefined.

$(2 x + 1) (x + 3) (x - 4) = 0$

Step 4

Solve for $x$ .

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Step 4.1

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

$2 x + 1 = 0$

$x + 3 = 0$

$x - 4 = 0$

Step 4.2

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 4.2.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 4.2.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 4.2.2.2

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

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Step 4.2.2.2.1

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

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

Step 4.2.2.2.2

Simplify the left side.

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Step 4.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.2.2.2.3

Simplify the right side.

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Step 4.2.2.2.3.1

Move the negative in front of the fraction.

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

Step 4.3

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

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Step 4.3.1

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

$x + 3 = 0$

Step 4.3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.4

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

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Step 4.4.1

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

$x - 4 = 0$

Step 4.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 4.5

The final solution is all the values that make $(2 x + 1) (x + 3) (x - 4) = 0$ true.

$x = - \frac{1}{2}, - 3, 4$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 3, x = - \frac{1}{2}, x = 4$

Step 6