Algebra Examples

$\frac{1}{2} + \frac{4}{2 x} = \frac{x + 4}{10}$

Step 1

Subtract $\frac{x + 4}{10}$ from both sides of the equation.

$\frac{1}{2} + \frac{4}{2 x} - \frac{x + 4}{10} = 0$

Step 2

Simplify $\frac{1}{2} + \frac{4}{2 x} - \frac{x + 4}{10}$ .

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Step 2.1

Cancel the common factor of $4$ and $2$ .

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Step 2.1.1

Factor $2$ out of $4$ .

$\frac{1}{2} + \frac{2 \cdot 2}{2 x} - \frac{x + 4}{10} = 0$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $2$ out of $2 x$ .

$\frac{1}{2} + \frac{2 \cdot 2}{2 (x)} - \frac{x + 4}{10} = 0$

Step 2.1.2.2

Cancel the common factor.

$\frac{1}{2} + \frac{2 \cdot 2}{2 x} - \frac{x + 4}{10} = 0$

Step 2.1.2.3

Rewrite the expression.

$\frac{1}{2} + \frac{2}{x} - \frac{x + 4}{10} = 0$

Step 2.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2}{x} + \frac{1}{2} \cdot \frac{5}{5} - \frac{x + 4}{10} = 0$

Step 2.3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1}{2}$ by $\frac{5}{5}$ .

$\frac{2}{x} + \frac{5}{2 \cdot 5} - \frac{x + 4}{10} = 0$

Step 2.3.2

Multiply $2$ by $5$ .

$\frac{2}{x} + \frac{5}{10} - \frac{x + 4}{10} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{2}{x} + \frac{5 - (x + 4)}{10} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{2}{x} + \frac{5 - x - 1 \cdot 4}{10} = 0$

Step 2.5.2

Multiply $- 1$ by $4$ .

$\frac{2}{x} + \frac{5 - x - 4}{10} = 0$

Step 2.5.3

Subtract $4$ from $5$ .

$\frac{2}{x} + \frac{- x + 1}{10} = 0$

Step 2.6

To write $\frac{2}{x}$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$\frac{2}{x} \cdot \frac{10}{10} + \frac{- x + 1}{10} = 0$

Step 2.7

To write $\frac{- x + 1}{10}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2}{x} \cdot \frac{10}{10} + \frac{- x + 1}{10} \cdot \frac{x}{x} = 0$

Step 2.8

Write each expression with a common denominator of $x \cdot 10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.8.1

Multiply $\frac{2}{x}$ by $\frac{10}{10}$ .

$\frac{2 \cdot 10}{x \cdot 10} + \frac{- x + 1}{10} \cdot \frac{x}{x} = 0$

Step 2.8.2

Multiply $\frac{- x + 1}{10}$ by $\frac{x}{x}$ .

$\frac{2 \cdot 10}{x \cdot 10} + \frac{(- x + 1) x}{10 x} = 0$

Step 2.8.3

Reorder the factors of $x \cdot 10$ .

$\frac{2 \cdot 10}{10 x} + \frac{(- x + 1) x}{10 x} = 0$

Step 2.9

Combine the numerators over the common denominator.

$\frac{2 \cdot 10 + (- x + 1) x}{10 x} = 0$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $2$ by $10$ .

$\frac{20 + (- x + 1) x}{10 x} = 0$

Step 2.10.2

Apply the distributive property.

$\frac{20 - x \cdot x + 1 x}{10 x} = 0$

Step 2.10.3

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

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Step 2.10.3.1

Move $x$ .

$\frac{20 - (x \cdot x) + 1 x}{10 x} = 0$

Step 2.10.3.2

Multiply $x$ by $x$ .

$\frac{20 - x^{2} + 1 x}{10 x} = 0$

Step 2.10.4

Multiply $x$ by $1$ .

$\frac{20 - x^{2} + x}{10 x} = 0$

Step 2.10.5

Reorder terms.

$\frac{- x^{2} + x + 20}{10 x} = 0$

Step 2.10.6

Factor by grouping.

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Step 2.10.6.1

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 $a \cdot c = - 1 \cdot 20 = - 20$ and whose sum is $b = 1$ .

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Step 2.10.6.1.1

Multiply by $1$ .

$\frac{- x^{2} + 1 x + 20}{10 x} = 0$

Step 2.10.6.1.2

Rewrite $1$ as $- 4$ plus $5$

$\frac{- x^{2} + (- 4 + 5) x + 20}{10 x} = 0$

Step 2.10.6.1.3

Apply the distributive property.

$\frac{- x^{2} - 4 x + 5 x + 20}{10 x} = 0$

Step 2.10.6.2

Factor out the greatest common factor from each group.

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Step 2.10.6.2.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 4 x) + 5 x + 20}{10 x} = 0$

Step 2.10.6.2.2

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

$\frac{x (- x - 4) - 5 (- x - 4)}{10 x} = 0$

Step 2.10.6.3

Factor the polynomial by factoring out the greatest common factor, $- x - 4$ .

$\frac{(- x - 4) (x - 5)}{10 x} = 0$

Step 2.11

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

$\frac{(- (x) - 4) (x - 5)}{10 x} = 0$

Step 2.12

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{(- (x) - 1 (4)) (x - 5)}{10 x} = 0$

Step 2.13

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

$\frac{- (x + 4) (x - 5)}{10 x} = 0$

Step 2.14

Rewrite $- (x + 4)$ as $- 1 (x + 4)$ .

$\frac{- 1 (x + 4) (x - 5)}{10 x} = 0$

Step 2.15

Move the negative in front of the fraction.

$- \frac{(x + 4) (x - 5)}{10 x} = 0$

Step 3

Set the numerator equal to zero.

$(x + 4) (x - 5) = 0$

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 $0$ .

$x + 4 = 0$

$x - 5 = 0$

Step 4.2

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

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Step 4.2.1

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

$x + 4 = 0$

Step 4.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4.3

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.4

The final solution is all the values that make $(x + 4) (x - 5) = 0$ true.

$x = - 4, 5$