Rational Expressions

mlbmaniaco
Right now in math class we are learning rational expressions. Since I am in an advanced math class, it seems like we learn a new lesson everyday. So if you don't understand something, you pretty much need to teach yourself. I don't really understand rational expressions, so can someone tell me if I am doing these two problems right(If I am doing wrong please tell me how to do them.)

Problem: x/3 = 4/x+4

Answer: 1) First I found a common denominator.
3x+12(x/3) = 3x+12(4/x+4)
2) So I got x+4 = x+3
3) Then the answer would be x=-4, x=-3

Am I right?

Homework Helper
Watch out with the brackets, I suppose you mean

$$\frac{x}{3} = \frac{4}{{x + 4}}$$

Now, multiply both sides with a common denominator to get rid of the denominators, so with for example $3\left( {x + 4} \right)$

Staff Emeritus
Gold Member
First off, you need to learn to use parentheses correctly. Whenever an addition or subtraction is supposed to happen before a multiplication or division, you need parentheses.

For example:

$$4/x+4 = \frac{4}{x} + 4$$

but

$$4/(x+4) = \frac{4}{x+4}$$

(3) is totally wrong -- if x+3 = x+4, then x=-4 and x=-3 are certainly not solutions.

But... (2) is also totally wrong. You skipped a bunch of steps, so I don't know what you're doing wrong. Multiplication by 3x+12 was a reasonable idea, though. Could you post your work?

mlbmaniaco
See, I don't know my work. I have no idea what I am doing. This is as far as I got with the book. I am supposed to simplify and check

Homework Helper
Well, try what I said. By multiplying both sides with 3, you lose the left denominator. Then, multiply both sides with x+4, this will get rid off the right denominator

mlbmaniaco
Another Question

so would i then have 3(3) = x2 = 4 ?

mlbmaniaco
Never Mind, I think I'll just give up. It is way to hard for me to understand

Homework Helper
I'll show you that first step. We multiply both sides with 3.
At the LHS, the 3 will cancel out with the denominator, as we wanted.
At the RHS, you can simplify it by multiplying it when the nominator.

$$\frac{x}{3} = \frac{4}{{x + 4}} \Leftrightarrow 3 \cdot \frac{x}{3} = 3 \cdot \frac{4}{{x + 4}} \Leftrightarrow x = \frac{{12}}{{x + 4}}$$

Now, try losing the right denominator by multiplying both sides with (x+4) in the same way

mlbmaniaco
So would I do this?

x+4 * x = 12/ x+4 * x+4

Then I would get . . .

x(x+4) = 12(x+4)

Right?

If so, what do I do next?

mlbmaniaco
wait I made a mistake . . .

It would be x(x=4) = 12
Right...

mlbmaniaco
I mean x(x+4) = 12

Right

Staff Emeritus
Gold Member
Yes:

x/3 = 4/(x+4)

implies

x(x+4) = 12

Homework Helper
mlbmaniaco said:
I mean x(x+4) = 12

Right
Correct!

Now bring everything to 1 side and you have a quadratic equation.
Solve with the quadratic formula or by factoring.

mlbmaniaco
We are supposed to solve by factoring, so how do I do that?

Homework Helper
So we have

$$x\left( {x + 4} \right) = 12 \Leftrightarrow x^2 + 4x - 12 = 0$$

Personally, I would factor just by finding zeroes
The divisors of the constant (-12) are 'possible candidates'...