Algebra Examples

$\frac{6 x}{x + 4} + 4 = \frac{2 x + 2}{x - 1}$

Step 1

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

$\frac{6 x}{x + 4} + 4 - \frac{2 x + 2}{x - 1} = 0$

Step 2

Simplify $\frac{6 x}{x + 4} + 4 - \frac{2 x + 2}{x - 1}$ .

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Step 2.1

Factor $2$ out of $2 x + 2$ .

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Step 2.1.1

Factor $2$ out of $2 x$ .

$\frac{6 x}{x + 4} + 4 - \frac{2 (x) + 2}{x - 1} = 0$

Step 2.1.2

Factor $2$ out of $2$ .

$\frac{6 x}{x + 4} + 4 - \frac{2 (x) + 2 (1)}{x - 1} = 0$

Step 2.1.3

Factor $2$ out of $2 (x) + 2 (1)$ .

$\frac{6 x}{x + 4} + 4 - \frac{2 (x + 1)}{x - 1} = 0$

Step 2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{6 x}{x + 4} + \frac{4 (x + 4)}{x + 4} - \frac{2 (x + 1)}{x - 1} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{6 x + 4 (x + 4)}{x + 4} - \frac{2 (x + 1)}{x - 1} = 0$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Factor $2$ out of $6 x + 4 (x + 4)$ .

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Step 2.4.1.1

Factor $2$ out of $6 x$ .

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

Step 2.4.1.2

Factor $2$ out of $4 (x + 4)$ .

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

Step 2.4.1.3

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

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

Step 2.4.2

Apply the distributive property.

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

Step 2.4.3

Multiply $2$ by $4$ .

$\frac{2 (3 x + 2 x + 8)}{x + 4} - \frac{2 (x + 1)}{x - 1} = 0$

Step 2.4.4

Add $3 x$ and $2 x$ .

$\frac{2 (5 x + 8)}{x + 4} - \frac{2 (x + 1)}{x - 1} = 0$

Step 2.5

To write $\frac{2 (5 x + 8)}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{2 (5 x + 8)}{x + 4} \cdot \frac{x - 1}{x - 1} - \frac{2 (x + 1)}{x - 1} = 0$

Step 2.6

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

$\frac{2 (5 x + 8)}{x + 4} \cdot \frac{x - 1}{x - 1} - \frac{2 (x + 1)}{x - 1} \cdot \frac{x + 4}{x + 4} = 0$

Step 2.7

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

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Step 2.7.1

Multiply $\frac{2 (5 x + 8)}{x + 4}$ by $\frac{x - 1}{x - 1}$ .

$\frac{2 (5 x + 8) (x - 1)}{(x + 4) (x - 1)} - \frac{2 (x + 1)}{x - 1} \cdot \frac{x + 4}{x + 4} = 0$

Step 2.7.2

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

$\frac{2 (5 x + 8) (x - 1)}{(x + 4) (x - 1)} - \frac{2 (x + 1) (x + 4)}{(x - 1) (x + 4)} = 0$

Step 2.7.3

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

$\frac{2 (5 x + 8) (x - 1)}{(x + 4) (x - 1)} - \frac{2 (x + 1) (x + 4)}{(x + 4) (x - 1)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{2 (5 x + 8) (x - 1) - 2 (x + 1) (x + 4)}{(x + 4) (x - 1)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Factor $2$ out of $2 (5 x + 8) (x - 1) - 2 (x + 1) (x + 4)$ .

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Step 2.9.1.1

Factor $2$ out of $2 (5 x + 8) (x - 1)$ .

$\frac{2 ((5 x + 8) (x - 1)) - 2 (x + 1) (x + 4)}{(x + 4) (x - 1)} = 0$

Step 2.9.1.2

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

$\frac{2 ((5 x + 8) (x - 1)) + 2 (- (x + 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.1.3

Factor $2$ out of $2 ((5 x + 8) (x - 1)) + 2 (- (x + 1) (x + 4))$ .

$\frac{2 ((5 x + 8) (x - 1) - (x + 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.2

Expand $(5 x + 8) (x - 1)$ using the FOIL Method.

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Step 2.9.2.1

Apply the distributive property.

$\frac{2 (5 x (x - 1) + 8 (x - 1) - (x + 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.2.2

Apply the distributive property.

$\frac{2 (5 x \cdot x + 5 x \cdot - 1 + 8 (x - 1) - (x + 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.2.3

Apply the distributive property.

$\frac{2 (5 x \cdot x + 5 x \cdot - 1 + 8 x + 8 \cdot - 1 - (x + 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.3

Simplify and combine like terms.

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Step 2.9.3.1

Simplify each term.

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Step 2.9.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.9.3.1.1.1

Move $x$ .

$\frac{2 (5 (x \cdot x) + 5 x \cdot - 1 + 8 x + 8 \cdot - 1 - (x + 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.3.1.1.2

Multiply $x$ by $x$ .

$\frac{2 (5 x^{2} + 5 x \cdot - 1 + 8 x + 8 \cdot - 1 - (x + 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.3.1.2

Multiply $- 1$ by $5$ .

$\frac{2 (5 x^{2} - 5 x + 8 x + 8 \cdot - 1 - (x + 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.3.1.3

Multiply $8$ by $- 1$ .

$\frac{2 (5 x^{2} - 5 x + 8 x - 8 - (x + 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.3.2

Add $- 5 x$ and $8 x$ .

$\frac{2 (5 x^{2} + 3 x - 8 - (x + 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.4

Apply the distributive property.

$\frac{2 (5 x^{2} + 3 x - 8 + (- x - 1 \cdot 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.5

Multiply $- 1$ by $1$ .

$\frac{2 (5 x^{2} + 3 x - 8 + (- x - 1) (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.6

Expand $(- x - 1) (x + 4)$ using the FOIL Method.

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Step 2.9.6.1

Apply the distributive property.

$\frac{2 (5 x^{2} + 3 x - 8 - x (x + 4) - 1 (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.6.2

Apply the distributive property.

$\frac{2 (5 x^{2} + 3 x - 8 - x \cdot x - x \cdot 4 - 1 (x + 4))}{(x + 4) (x - 1)} = 0$

Step 2.9.6.3

Apply the distributive property.

$\frac{2 (5 x^{2} + 3 x - 8 - x \cdot x - x \cdot 4 - 1 x - 1 \cdot 4)}{(x + 4) (x - 1)} = 0$

Step 2.9.7

Simplify and combine like terms.

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Step 2.9.7.1

Simplify each term.

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Step 2.9.7.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.9.7.1.1.1

Move $x$ .

$\frac{2 (5 x^{2} + 3 x - 8 - (x \cdot x) - x \cdot 4 - 1 x - 1 \cdot 4)}{(x + 4) (x - 1)} = 0$

Step 2.9.7.1.1.2

Multiply $x$ by $x$ .

$\frac{2 (5 x^{2} + 3 x - 8 - x^{2} - x \cdot 4 - 1 x - 1 \cdot 4)}{(x + 4) (x - 1)} = 0$

Step 2.9.7.1.2

Multiply $4$ by $- 1$ .

$\frac{2 (5 x^{2} + 3 x - 8 - x^{2} - 4 x - 1 x - 1 \cdot 4)}{(x + 4) (x - 1)} = 0$

Step 2.9.7.1.3

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

$\frac{2 (5 x^{2} + 3 x - 8 - x^{2} - 4 x - x - 1 \cdot 4)}{(x + 4) (x - 1)} = 0$

Step 2.9.7.1.4

Multiply $- 1$ by $4$ .

$\frac{2 (5 x^{2} + 3 x - 8 - x^{2} - 4 x - x - 4)}{(x + 4) (x - 1)} = 0$

Step 2.9.7.2

Subtract $x$ from $- 4 x$ .

$\frac{2 (5 x^{2} + 3 x - 8 - x^{2} - 5 x - 4)}{(x + 4) (x - 1)} = 0$

Step 2.9.8

Subtract $x^{2}$ from $5 x^{2}$ .

$\frac{2 (4 x^{2} + 3 x - 8 - 5 x - 4)}{(x + 4) (x - 1)} = 0$

Step 2.9.9

Subtract $5 x$ from $3 x$ .

$\frac{2 (4 x^{2} - 2 x - 8 - 4)}{(x + 4) (x - 1)} = 0$

Step 2.9.10

Subtract $4$ from $- 8$ .

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

Step 2.9.11

Rewrite $4 x^{2} - 2 x - 12$ in a factored form.

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Step 2.9.11.1

Factor $2$ out of $4 x^{2} - 2 x - 12$ .

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Step 2.9.11.1.1

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

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

Step 2.9.11.1.2

Factor $2$ out of $- 2 x$ .

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

Step 2.9.11.1.3

Factor $2$ out of $- 12$ .

$\frac{2 (2 (2 x^{2}) + 2 (- x) + 2 (- 6))}{(x + 4) (x - 1)} = 0$

Step 2.9.11.1.4

Factor $2$ out of $2 (2 x^{2}) + 2 (- x)$ .

$\frac{2 (2 (2 x^{2} - x) + 2 (- 6))}{(x + 4) (x - 1)} = 0$

Step 2.9.11.1.5

Factor $2$ out of $2 (2 x^{2} - x) + 2 (- 6)$ .

$\frac{2 (2 (2 x^{2} - x - 6))}{(x + 4) (x - 1)} = 0$

Step 2.9.11.2

Factor by grouping.

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Step 2.9.11.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = - 1$ .

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Step 2.9.11.2.1.1

Factor $- 1$ out of $- x$ .

$\frac{2 (2 (2 x^{2} - (x) - 6))}{(x + 4) (x - 1)} = 0$

Step 2.9.11.2.1.2

Rewrite $- 1$ as $3$ plus $- 4$

$\frac{2 (2 (2 x^{2} + (3 - 4) x - 6))}{(x + 4) (x - 1)} = 0$

Step 2.9.11.2.1.3

Apply the distributive property.

$\frac{2 (2 (2 x^{2} + 3 x - 4 x - 6))}{(x + 4) (x - 1)} = 0$

Step 2.9.11.2.2

Factor out the greatest common factor from each group.

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Step 2.9.11.2.2.1

Group the first two terms and the last two terms.

$\frac{2 (2 ((2 x^{2} + 3 x) - 4 x - 6))}{(x + 4) (x - 1)} = 0$

Step 2.9.11.2.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 2.9.11.2.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

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

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

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

Step 2.9.12

Multiply $2$ by $2$ .

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

Step 3

Set the numerator equal to zero.

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

Step 4

Solve the equation 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 $0$ .

$2 x + 3 = 0$

$x - 2 = 0$

Step 4.2

Set $2 x + 3$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Step 4.2.2

Solve $2 x + 3 = 0$ for $x$ .

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Step 4.2.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 4.2.2.2

Divide each term in $2 x = - 3$ by $2$ and simplify.

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Step 4.2.2.2.1

Divide each term in $2 x = - 3$ by $2$ .

$\frac{2 x}{2} = \frac{- 3}{2}$

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.

$\frac{2 x}{2} = \frac{- 3}{2}$

Step 4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{2}$

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.

$x = - \frac{3}{2}$

Step 4.3

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.4

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

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