Precalculus Examples

$\frac{10}{x + 4} = \frac{15}{4 (x + 1)}$

Step 1

Subtract $\frac{15}{4 (x + 1)}$ from both sides of the equation.

$\frac{10}{x + 4} - \frac{15}{4 (x + 1)} = 0$

Step 2

Simplify $\frac{10}{x + 4} - \frac{15}{4 (x + 1)}$ .

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Step 2.1

To write $\frac{10}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{4 (x + 1)}{4 (x + 1)}$ .

$\frac{10}{x + 4} \cdot \frac{4 (x + 1)}{4 (x + 1)} - \frac{15}{4 (x + 1)} = 0$

Step 2.2

To write $- \frac{15}{4 (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{10}{x + 4} \cdot \frac{4 (x + 1)}{4 (x + 1)} - \frac{15}{4 (x + 1)} \cdot \frac{x + 4}{x + 4} = 0$

Step 2.3

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

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Step 2.3.1

Multiply $\frac{10}{x + 4}$ by $\frac{4 (x + 1)}{4 (x + 1)}$ .

$\frac{10 (4 (x + 1))}{(x + 4) (4 (x + 1))} - \frac{15}{4 (x + 1)} \cdot \frac{x + 4}{x + 4} = 0$

Step 2.3.2

Multiply $\frac{15}{4 (x + 1)}$ by $\frac{x + 4}{x + 4}$ .

$\frac{10 (4 (x + 1))}{(x + 4) (4 (x + 1))} - \frac{15 (x + 4)}{4 (x + 1) (x + 4)} = 0$

Step 2.3.3

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

$\frac{10 (4 (x + 1))}{4 (x + 1) (x + 4)} - \frac{15 (x + 4)}{4 (x + 1) (x + 4)} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{10 (4 (x + 1)) - 15 (x + 4)}{4 (x + 1) (x + 4)} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $5$ out of $10 \cdot 4 (x + 1) - 15 (x + 4)$ .

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Step 2.5.1.1

Factor $5$ out of $10 \cdot 4 (x + 1)$ .

$\frac{5 (2 \cdot 4 (x + 1)) - 15 (x + 4)}{4 (x + 1) (x + 4)} = 0$

Step 2.5.1.2

Factor $5$ out of $- 15 (x + 4)$ .

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

Step 2.5.1.3

Factor $5$ out of $5 (2 \cdot 4 (x + 1)) + 5 (- 3 (x + 4))$ .

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

Step 2.5.2

Multiply $2$ by $4$ .

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

Step 2.5.3

Apply the distributive property.

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

Step 2.5.4

Multiply $8$ by $1$ .

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

Step 2.5.5

Apply the distributive property.

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

Step 2.5.6

Multiply $- 3$ by $4$ .

$\frac{5 (8 x + 8 - 3 x - 12)}{4 (x + 1) (x + 4)} = 0$

Step 2.5.7

Subtract $3 x$ from $8 x$ .

$\frac{5 (5 x + 8 - 12)}{4 (x + 1) (x + 4)} = 0$

Step 2.5.8

Subtract $12$ from $8$ .

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

Step 3

Set the numerator equal to zero.

$5 (5 x - 4) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $5 (5 x - 4) = 0$ by $5$ and simplify.

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Step 4.1.1

Divide each term in $5 (5 x - 4) = 0$ by $5$ .

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

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $5$ .

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Step 4.1.2.1.1

Cancel the common factor.

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

Step 4.1.2.1.2

Divide $5 x - 4$ by $1$ .

$5 x - 4 = \frac{0}{5}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $5$ .

$5 x - 4 = 0$

Step 4.2

Add $4$ to both sides of the equation.

$5 x = 4$

Step 4.3

Divide each term in $5 x = 4$ by $5$ and simplify.

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Step 4.3.1

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $5$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{4}{5}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{5}$