Algebra Examples

$\frac{11 (x - 3)}{x - 4} + \frac{5}{x} = \frac{- 5}{x (x - 4)}$

Step 1

Subtract $\frac{- 5}{x (x - 4)}$ from both sides of the equation.

$\frac{11 (x - 3)}{x - 4} + \frac{5}{x} - \frac{- 5}{x (x - 4)} = 0$

Step 2

Simplify $\frac{11 (x - 3)}{x - 4} + \frac{5}{x} - \frac{- 5}{x (x - 4)}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Move the negative in front of the fraction.

$\frac{11 (x - 3)}{x - 4} + \frac{5}{x} - - \frac{5}{x (x - 4)} = 0$

Step 2.1.2

Multiply $- - \frac{5}{x (x - 4)}$ .

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Step 2.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{11 (x - 3)}{x - 4} + \frac{5}{x} + 1 \frac{5}{x (x - 4)} = 0$

Step 2.1.2.2

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

$\frac{11 (x - 3)}{x - 4} + \frac{5}{x} + \frac{5}{x (x - 4)} = 0$

Step 2.2

To write $\frac{11 (x - 3)}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{11 (x - 3)}{x - 4} \cdot \frac{x}{x} + \frac{5}{x} + \frac{5}{x (x - 4)} = 0$

Step 2.3

To write $\frac{5}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{11 (x - 3)}{x - 4} \cdot \frac{x}{x} + \frac{5}{x} \cdot \frac{x - 4}{x - 4} + \frac{5}{x (x - 4)} = 0$

Step 2.4

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

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Step 2.4.1

Multiply $\frac{11 (x - 3)}{x - 4}$ by $\frac{x}{x}$ .

$\frac{11 (x - 3) x}{(x - 4) x} + \frac{5}{x} \cdot \frac{x - 4}{x - 4} + \frac{5}{x (x - 4)} = 0$

Step 2.4.2

Multiply $\frac{5}{x}$ by $\frac{x - 4}{x - 4}$ .

$\frac{11 (x - 3) x}{(x - 4) x} + \frac{5 (x - 4)}{x (x - 4)} + \frac{5}{x (x - 4)} = 0$

Step 2.4.3

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

$\frac{11 (x - 3) x}{x (x - 4)} + \frac{5 (x - 4)}{x (x - 4)} + \frac{5}{x (x - 4)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{11 (x - 3) x + 5 (x - 4)}{x (x - 4)} + \frac{5}{x (x - 4)} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{(11 x + 11 \cdot - 3) x + 5 (x - 4)}{x (x - 4)} + \frac{5}{x (x - 4)} = 0$

Step 2.6.2

Multiply $11$ by $- 3$ .

$\frac{(11 x - 33) x + 5 (x - 4)}{x (x - 4)} + \frac{5}{x (x - 4)} = 0$

Step 2.6.3

Apply the distributive property.

$\frac{11 x \cdot x - 33 x + 5 (x - 4)}{x (x - 4)} + \frac{5}{x (x - 4)} = 0$

Step 2.6.4

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

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Step 2.6.4.1

Move $x$ .

$\frac{11 (x \cdot x) - 33 x + 5 (x - 4)}{x (x - 4)} + \frac{5}{x (x - 4)} = 0$

Step 2.6.4.2

Multiply $x$ by $x$ .

$\frac{11 x^{2} - 33 x + 5 (x - 4)}{x (x - 4)} + \frac{5}{x (x - 4)} = 0$

Step 2.6.5

Apply the distributive property.

$\frac{11 x^{2} - 33 x + 5 x + 5 \cdot - 4}{x (x - 4)} + \frac{5}{x (x - 4)} = 0$

Step 2.6.6

Multiply $5$ by $- 4$ .

$\frac{11 x^{2} - 33 x + 5 x - 20}{x (x - 4)} + \frac{5}{x (x - 4)} = 0$

Step 2.6.7

Add $- 33 x$ and $5 x$ .

$\frac{11 x^{2} - 28 x - 20}{x (x - 4)} + \frac{5}{x (x - 4)} = 0$

Step 2.7

Combine the numerators over the common denominator.

$\frac{11 x^{2} - 28 x - 20 + 5}{x (x - 4)} = 0$

Step 2.8

Add $- 20$ and $5$ .

$\frac{11 x^{2} - 28 x - 15}{x (x - 4)} = 0$

Step 2.9

Factor by grouping.

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Step 2.9.1

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 $a \cdot c = 11 \cdot - 15 = - 165$ and whose sum is $b = - 28$ .

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Step 2.9.1.1

Factor $- 28$ out of $- 28 x$ .

$\frac{11 x^{2} - 28 (x) - 15}{x (x - 4)} = 0$

Step 2.9.1.2

Rewrite $- 28$ as $5$ plus $- 33$

$\frac{11 x^{2} + (5 - 33) x - 15}{x (x - 4)} = 0$

Step 2.9.1.3

Apply the distributive property.

$\frac{11 x^{2} + 5 x - 33 x - 15}{x (x - 4)} = 0$

Step 2.9.2

Factor out the greatest common factor from each group.

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Step 2.9.2.1

Group the first two terms and the last two terms.

$\frac{(11 x^{2} + 5 x) - 33 x - 15}{x (x - 4)} = 0$

Step 2.9.2.2

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

$\frac{x (11 x + 5) - 3 (11 x + 5)}{x (x - 4)} = 0$

Step 2.9.3

Factor the polynomial by factoring out the greatest common factor, $11 x + 5$ .

$\frac{(11 x + 5) (x - 3)}{x (x - 4)} = 0$

Step 3

Set the numerator equal to zero.

$(11 x + 5) (x - 3) = 0$

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 $0$ .

$11 x + 5 = 0$

$x - 3 = 0$

Step 4.2

Set $11 x + 5$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $11 x + 5$ equal to $0$ .

$11 x + 5 = 0$

Step 4.2.2

Solve $11 x + 5 = 0$ for $x$ .

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Step 4.2.2.1

Subtract $5$ from both sides of the equation.

$11 x = - 5$

Step 4.2.2.2

Divide each term in $11 x = - 5$ by $11$ and simplify.

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Step 4.2.2.2.1

Divide each term in $11 x = - 5$ by $11$ .

$\frac{11 x}{11} = \frac{- 5}{11}$

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 $11$ .

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Step 4.2.2.2.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{- 5}{11}$

Step 4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{11}$

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.

$x = - \frac{5}{11}$

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

Add $3$ to both sides of the equation.

$x = 3$

Step 4.4

The final solution is all the values that make $(11 x + 5) (x - 3) = 0$ true.

$x = - \frac{5}{11}, 3$