This is a difference of squares. The difference of squares formula is _a_2 – _b_2 = (a + b)(a – b).

In this problem, a = 6_x_ and b = 7_y_:

36_x_2 – 49_y_2 = (6_x_ + 7_y_)(6_x_ – 7_y_)

Multiply both sides by :

Factor this using the -method. We split the middle term using two integers whose sum is and whose product is . These integers are :

Set each factor equal to 0 and solve separately:

or

Factor:

Double check by factoring:

Add together:

Therefore:

Start by setting the inequality to zero and by solving for .

Now, plot these two points on to a number line.

Notice that these two numbers effectively divide up the number line into three regions:

, , and

Now, choose a number in each of these regions and put it back in the factored inequality to see which cases are true.

For , let

Since this is not less than , the solution to this inequality cannot lie in this region.

For , let .

Since this will make the inequality true, the solution can lie in this region.

Finally, for , let

Since this number is not less than zero, the solution cannot lie in this region.

Thus, the solution to this inequality is

To multiple these binomials, you can use the FOIL method to multiply each of the expressions individually.This will give you

or .

Determine the least common denominator in order to add the numerator.

Each denominator shares an term. The least common denominator is since it is divisible by each coefficient of the denominator.

Convert the fractions.

Simplify the top and bottom.

The answer is:

The two fractions on the left side of the equation need a common denominator. We can easily do find one by multiplying both the top and bottom of each fraction by the denominator of the other.

becomes .

becomes .

Now add the two fractions:

To solve, multiply both sides of the equation by , yielding

.

Multiply both sides by 3:

Move all terms to the same side:

This looks like a complicated equation to factor, but luckily, the only factors of 37 are 37 and 1, so we are left with

.

Our solutions are therefore

and

.

Rewrite the left fraction using common factors.

Cancel out common terms.

Factorize the bottom term.

The answer is:

This is a difference of squares. The difference of squares formula is _a_2 – _b_2 = (a + b)(a – b).

In this problem, a = 6_x_ and b = 7_y_:

36_x_2 – 49_y_2 = (6_x_ + 7_y_)(6_x_ – 7_y_)

In order to simplify the rational expression, we will need to rewrite the expression as a multiplication sign and take the reciprocal of the second term.

Simplify the numerator.

Simplify the denominator by FOIL method.

Divide the numerator with the denominator.

The answer is: