Pre-Algebra Examples

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

Step 1

Simplify $\frac{x}{3} - \frac{5}{x + 2}$ .

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Step 1.1

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

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

Step 1.2

To write $- \frac{5}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

Multiply $\frac{5}{x + 2}$ by $\frac{3}{3}$ .

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

Step 1.3.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Multiply $x$ by $x$ .

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

Step 1.5.3

Move $2$ to the left of $x$ .

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

Step 1.5.4

Multiply $- 5$ by $3$ .

$\frac{x^{2} + 2 x - 15}{3 (x + 2)} = 0$

Step 1.5.5

Factor $x^{2} + 2 x - 15$ using the AC method.

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Step 1.5.5.1

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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 1.5.5.2

Write the factored form using these integers.

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

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = - 5, 3$

Step 3