Algebra Examples

$\frac{4 x}{x - 3} + \frac{x}{2} = \frac{12}{x - 3}$

Step 1

Subtract $\frac{12}{x - 3}$ from both sides of the equation.

$\frac{4 x}{x - 3} + \frac{x}{2} - \frac{12}{x - 3} = 0$

Step 2

Simplify $\frac{4 x}{x - 3} + \frac{x}{2} - \frac{12}{x - 3}$ .

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Step 2.1

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

$\frac{4 x}{x - 3} \cdot \frac{2}{2} + \frac{x}{2} - \frac{12}{x - 3} = 0$

Step 2.2

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

$\frac{4 x}{x - 3} \cdot \frac{2}{2} + \frac{x}{2} \cdot \frac{x - 3}{x - 3} - \frac{12}{x - 3} = 0$

Step 2.3

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

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Step 2.3.1

Multiply $\frac{4 x}{x - 3}$ by $\frac{2}{2}$ .

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

Step 2.3.2

Multiply $\frac{x}{2}$ by $\frac{x - 3}{x - 3}$ .

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

Step 2.3.3

Reorder the factors of $(x - 3) \cdot 2$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

Factor $x$ out of $4 x \cdot 2$ .

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

Step 2.5.1.2

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

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

Step 2.5.2

Multiply $4$ by $2$ .

$\frac{x (8 + x - 3)}{2 (x - 3)} - \frac{12}{x - 3} = 0$

Step 2.5.3

Subtract $3$ from $8$ .

$\frac{x (x + 5)}{2 (x - 3)} - \frac{12}{x - 3} = 0$

Step 2.6

To write $- \frac{12}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x (x + 5)}{2 (x - 3)} - \frac{12}{x - 3} \cdot \frac{2}{2} = 0$

Step 2.7

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

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Step 2.7.1

Multiply $\frac{12}{x - 3}$ by $\frac{2}{2}$ .

$\frac{x (x + 5)}{2 (x - 3)} - \frac{12 \cdot 2}{(x - 3) \cdot 2} = 0$

Step 2.7.2

Reorder the factors of $(x - 3) \cdot 2$ .

$\frac{x (x + 5)}{2 (x - 3)} - \frac{12 \cdot 2}{2 (x - 3)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{x (x + 5) - 12 \cdot 2}{2 (x - 3)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 5 - 12 \cdot 2}{2 (x - 3)} = 0$

Step 2.9.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 5 - 12 \cdot 2}{2 (x - 3)} = 0$

Step 2.9.3

Move $5$ to the left of $x$ .

$\frac{x^{2} + 5 \cdot x - 12 \cdot 2}{2 (x - 3)} = 0$

Step 2.9.4

Multiply $- 12$ by $2$ .

$\frac{x^{2} + 5 x - 24}{2 (x - 3)} = 0$

Step 2.9.5

Factor $x^{2} + 5 x - 24$ using the AC method.

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Step 2.9.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $5$ .

$- 3, 8$

Step 2.9.5.2

Write the factored form using these integers.

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

Step 2.10

Cancel the common factor of $x - 3$ .

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Step 2.10.1

Cancel the common factor.

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

Step 2.10.2

Rewrite the expression.

$\frac{x + 8}{2} = 0$

Step 3

Set the numerator equal to zero.

$x + 8 = 0$

Step 4

Subtract $8$ from both sides of the equation.

$x = - 8$