Algebra Examples

$\frac{6}{x} + \frac{x - 3}{4} = 2$

Step 1

Subtract $2$ from both sides of the equation.

$\frac{6}{x} + \frac{x - 3}{4} - 2 = 0$

Step 2

Simplify $\frac{6}{x} + \frac{x - 3}{4} - 2$ .

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Step 2.1

To write $\frac{6}{x}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{6}{x} \cdot \frac{4}{4} + \frac{x - 3}{4} - 2 = 0$

Step 2.2

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

$\frac{6}{x} \cdot \frac{4}{4} + \frac{x - 3}{4} \cdot \frac{x}{x} - 2 = 0$

Step 2.3

Write each expression with a common denominator of $x \cdot 4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{6}{x}$ by $\frac{4}{4}$ .

$\frac{6 \cdot 4}{x \cdot 4} + \frac{x - 3}{4} \cdot \frac{x}{x} - 2 = 0$

Step 2.3.2

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

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

Step 2.3.3

Reorder the factors of $x \cdot 4$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Multiply $6$ by $4$ .

$\frac{24 + (x - 3) x}{4 x} - 2 = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{24 + x \cdot x - 3 x}{4 x} - 2 = 0$

Step 2.6.2

Multiply $x$ by $x$ .

$\frac{24 + x^{2} - 3 x}{4 x} - 2 = 0$

Step 2.6.3

Reorder terms.

$\frac{x^{2} - 3 x + 24}{4 x} - 2 = 0$

Step 2.7

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

$\frac{x^{2} - 3 x + 24}{4 x} - 2 \cdot \frac{4 x}{4 x} = 0$

Step 2.8

Combine $- 2$ and $\frac{4 x}{4 x}$ .

$\frac{x^{2} - 3 x + 24}{4 x} + \frac{- 2 (4 x)}{4 x} = 0$

Step 2.9

Combine the numerators over the common denominator.

$\frac{x^{2} - 3 x + 24 - 2 (4 x)}{4 x} = 0$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 2$ by $4$ .

$\frac{x^{2} - 3 x + 24 - 8 x}{4 x} = 0$

Step 2.10.2

Subtract $8 x$ from $- 3 x$ .

$\frac{x^{2} - 11 x + 24}{4 x} = 0$

Step 2.10.3

Factor $x^{2} - 11 x + 24$ using the AC method.

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Step 2.10.3.1

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 $- 11$ .

$- 8, - 3$

Step 2.10.3.2

Write the factored form using these integers.

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

Step 3

Set the numerator equal to zero.

$(x - 8) (x - 3) = 0$

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 $0$ .

$x - 8 = 0$

$x - 3 = 0$

Step 4.2

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 4.2.2

Add $8$ to both sides of the equation.

$x = 8$

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 $(x - 8) (x - 3) = 0$ true.

$x = 8, 3$