Solving Equations Using Rational Expressions: Simplify x/(x+3) + (9x+18)/(x^2+3x)

Dirty.Rotten.Imbecile said:
View attachment 9526

The text says says the answer is x+6/x but I can’t figure out how they came up with that answer.

can anyone help?
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What you did is similar to saying 1/5 + 1/6 = 30(1/5) + 30(1/6) = 6 + 5 = 11. So in the end, 1/5 + 1/6 = 11. Do you believe that????

What went wrong is 1/5 \(\displaystyle \neq\) 30(1/5) = 6 and 1/6 \(\displaystyle \neq\) 30(1/6) = 5.

Rather you should get common denominators which is x(x+3). That is for the left fraction multiply by x/x which equals 1, so you are NOT changing the value. The right hand fraction actually has the denominator you want so do not change in. Factor the numerator and reduce.

Note that you can multiply by x(x+3) IF you do the same to both sides. In your case you only had one side.
 
Dirty.Rotten.Imbecile said:
The text says says the answer is (x+6)/x but I can’t figure out how they came up with that answer.
Click to expand...

Here is the straightforward answer: Since you multiplied the expression by x(x+3), you changed the value. But the goal is to rewrite the expression as a simpler, equivalent expression -- it has to have the same value. To accomplish that goal, you have to also divide by x(x+3) -- that is, put that as the denominator.

So you end up with (x2+9x+18)/[x(x+3)]. Now just simplify: Factor the numerator, and cancel. You'll end up with (x+6)/x.

The usual procedure is to multiply and divide at the same time: multiply the first fraction by x/x, giving it the same denominator as the second, and then add the numerators together. Check your textbook, which should have some examples of the process.
 
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