The number is_____?

Question

Six divided by the difference of a number and 3​, minus 5 divided by a number plus 3​, equals 4 times the reciprocal of the difference of the number squared and 9. What is the​ number?

Answer 1

Writing this one out in maths terms, using "x" in place of the phrase "number", and brackets and square brackets to separate information (I'm going off of the commas in your question):

[6/(x - 3)] - [5/(x + 3)] = 4 * [1/(x^2 - 9)]

How to come up with this equation?:

"Difference of a number and 3" just means "x - 3"

"Reciprocal of A" just means "1/A"

Now that everything is in maths terms, it's possible to solve for x:

[6/(x - 3)] - [5/(x + 3)] = 4 * [1/(x² - 9)]

We want to get rid of the xs in the denominators (on the bottom), so multiplying both sides by (x-3) gives:

6 - [5*(x - 3)]/(x+3) = 4*(x - 3) / (x²- 9)

Then distributing the four on the right side, then multiplying both sides by (x+3) gives:

6*(x+3) - [5*(x - 3)] = [(4x - 12)*(x+3)] / (x²- 9)

Distributing the 5 and 6 on the left side, and distributing the "-" before the 5 on the left side, and using FOIL rule on the right:

6x+ 18 - 5x + 15 = (4x² - 12x + 12x - 36)/ (x²- 9)

Simplifying:

(x + 33) = (4x² - 36)/(x²- 9)

Here, we can see that the numerator on the right side can be rewritten:

(x+33) = [4*(x² - 9)] / (x² - 9)

And this cancels out on the right side, giving:

(x+33) = 4

x = 4 - 33

x = -29

Answer 2

the number is -29