Simplify ((3x+15)/(49x^2-4))÷((2x-7)/(7x-2))

$\frac{3 x + 15}{49 x^{2} - 4} \div \frac{2 x - 7}{7 x - 2}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x + 15}{49 x^{2} - 4} \cdot \frac{7 x - 2}{2 x - 7}$

Step 2

Factor $3$ out of $3 x + 15$ .

Factor $3$ out of $3 x$ .

$\frac{3 (x) + 15}{49 x^{2} - 4} \cdot \frac{7 x - 2}{2 x - 7}$

Factor $3$ out of $15$ .

$\frac{3 x + 3 \cdot 5}{49 x^{2} - 4} \cdot \frac{7 x - 2}{2 x - 7}$

Factor $3$ out of $3 x + 3 \cdot 5$ .

$\frac{3 (x + 5)}{49 x^{2} - 4} \cdot \frac{7 x - 2}{2 x - 7}$

Step 3

Simplify the denominator.

Rewrite $49 x^{2}$ as ${(7 x)}^{2}$ .

$\frac{3 (x + 5)}{{(7 x)}^{2} - 4} \cdot \frac{7 x - 2}{2 x - 7}$

Rewrite $4$ as $2^{2}$ .

$\frac{3 (x + 5)}{{(7 x)}^{2} - 2^{2}} \cdot \frac{7 x - 2}{2 x - 7}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7 x$ and $b = 2$ .

$\frac{3 (x + 5)}{(7 x + 2) (7 x - 2)} \cdot \frac{7 x - 2}{2 x - 7}$

Step 4

Simplify terms.

Cancel the common factor of $7 x - 2$ .

Factor $7 x - 2$ out of $(7 x + 2) (7 x - 2)$ .

$\frac{3 (x + 5)}{(7 x - 2) (7 x + 2)} \cdot \frac{7 x - 2}{2 x - 7}$

Cancel the common factor.

$\frac{3 (x + 5)}{(7 x - 2) (7 x + 2)} \cdot \frac{7 x - 2}{2 x - 7}$

Rewrite the expression.

$\frac{3 (x + 5)}{7 x + 2} \cdot \frac{1}{2 x - 7}$

Multiply $\frac{3 (x + 5)}{7 x + 2}$ by $\frac{1}{2 x - 7}$ .

$\frac{3 (x + 5)}{(7 x + 2) (2 x - 7)}$