Algebra Examples

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

Step 1

To write $\frac{3 x}{2 x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

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

Step 2

To write $- \frac{3}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{2 x - 3}{2 x - 3}$ .

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

Step 3

Write each expression with a common denominator of $(2 x - 3) (x + 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{3 x}{2 x - 3}$ by $\frac{x + 6}{x + 6}$ .

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

Step 3.2

Multiply $\frac{3}{x + 6}$ by $\frac{2 x - 3}{2 x - 3}$ .

$\frac{3 x (x + 6)}{(2 x - 3) (x + 6)} - \frac{3 (2 x - 3)}{(x + 6) (2 x - 3)}$

Step 3.3

Reorder the factors of $(x + 6) (2 x - 3)$ .

$\frac{3 x (x + 6)}{(2 x - 3) (x + 6)} - \frac{3 (2 x - 3)}{(2 x - 3) (x + 6)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 x (x + 6) - 3 (2 x - 3)}{(2 x - 3) (x + 6)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $3$ out of $3 x (x + 6) - 3 (2 x - 3)$ .

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Step 5.1.1

Factor $3$ out of $3 x (x + 6)$ .

$\frac{3 (x (x + 6)) - 3 (2 x - 3)}{(2 x - 3) (x + 6)}$

Step 5.1.2

Factor $3$ out of $- 3 (2 x - 3)$ .

$\frac{3 (x (x + 6)) + 3 (- (2 x - 3))}{(2 x - 3) (x + 6)}$

Step 5.1.3

Factor $3$ out of $3 (x (x + 6)) + 3 (- (2 x - 3))$ .

$\frac{3 (x (x + 6) - (2 x - 3))}{(2 x - 3) (x + 6)}$

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $x$ by $x$ .

$\frac{3 (x^{2} + x \cdot 6 - (2 x - 3))}{(2 x - 3) (x + 6)}$

Step 5.4

Move $6$ to the left of $x$ .

$\frac{3 (x^{2} + 6 \cdot x - (2 x - 3))}{(2 x - 3) (x + 6)}$

Step 5.5

Apply the distributive property.

$\frac{3 (x^{2} + 6 x - (2 x) - - 3)}{(2 x - 3) (x + 6)}$

Step 5.6

Multiply $2$ by $- 1$ .

$\frac{3 (x^{2} + 6 x - 2 x - - 3)}{(2 x - 3) (x + 6)}$

Step 5.7

Multiply $- 1$ by $- 3$ .

$\frac{3 (x^{2} + 6 x - 2 x + 3)}{(2 x - 3) (x + 6)}$

Step 5.8

Subtract $2 x$ from $6 x$ .

$\frac{3 (x^{2} + 4 x + 3)}{(2 x - 3) (x + 6)}$

Step 5.9

Factor $x^{2} + 4 x + 3$ using the AC method.

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Step 5.9.1

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 $3$ and whose sum is $4$ .

$1, 3$

Step 5.9.2

Write the factored form using these integers.

$\frac{3 ((x + 1) (x + 3))}{(2 x - 3) (x + 6)}$

$\frac{3 (x + 1) (x + 3)}{(2 x - 3) (x + 6)}$