SOLUTION: Solve the equation (4w-1)/(3w+6)-(w-1)/3=(w-1)/w+2

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Question 121085: Solve the equation (4w-1)/(3w+6)-(w-1)/3=(w-1)/w+2
Found 2 solutions by solver91311, jim_thompson5910:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
%284w-1%29%2F%283w%2B6%29-%28w-1%29%2F3=%28w-1%29%2F%28w%2B2%29

Step 1 is to put your variable on one side, so add -%28%28w-1%29%2F%28w%2B2%29%29 to both sides of the equation

%284w-1%29%2F%283w%2B6%29-%28w-1%29%2F3-%28w-1%29%2F%28w%2B2%29=0

Now we need a common denominator. Notice that 3w%2B6 factors to 3%2A%28w%2B2%29 and those two factors are the other two denominators. Therefore, 3w%2B6 is our lowest common denominator.

We can leave the first term alone, but the second term has to be multiplied by %28w%2B2%29%2F%28w%2B2%29 and the third term must be multiplied by 3%2F3



Now is a good time to note that since w%2B2 is a factor in the denominator, we need to exclude -2 from the solution set because that would make a denominator or denominators in the original equation go to zero.

Multiply the binomial factors and distribute the 3

%284w-1-w%5E2-w%2B2-3w%2B3%29%2F%283w%2B6%29=0

Collect terms

4-w%5E2=0

Factor the difference of two squares:

%282-w%29%282%2Bw%29=0

So, 2-w=0 or 2%2Bw=0, in other words, w=2 or w=-2. But remember, we have to exclude -2 from the solution set because that value makes the original equation undefined. In the process of applying the common denominator, we introduced the square of the variable and that process introduced an extraneous root.

x=2 is the only root of the equation.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

%284w-1%29%2F%283w%2B6%29-%28w-1%29%2F%283%29=%28w-1%29%2F%28w%2B2%29 Start with the given equation


Multiply both sides by the LCD %283w%2B6%29%283%29%28w%2B2%29. Doing this will eliminate every fraction.


%283%29%28w%2B2%29%284w-1%29-%283w%2B6%29%28w%2B2%29%28w-1%29=%283w%2B6%29%283%29%28w-1%29 Distribute and multiply. Notice every denominator has been canceled out.



%283%29%284w%5E2%2B7w-2%29-%283w%2B6%29%28w%5E2%2Bw-2%29=3%283w%5E2%2B3w-6%29 Foil



%283%29%284w%5E2%2B7w-2%29-%283w%5E3%2B9w%5E2-12%29=3%283w%5E2%2B3w-6%29 Multiply 3w%2B6 and w%5E2%2Bw-2 to get 3w%5E3%2B9w%5E2-12



12w%5E2%2B21w-6-3w%5E3-9w%5E2%2B12=9w%5E2%2B9w-18 Distribute



-3w%5E3-6w%5E2%2B12w%2B24=0 Get everything to the left side


-3%28w%5E2-4%29%28w%2B2%29=0 Factor the left side


Now set each factor equal to zero:

w%5E2-4=0 or w%2B2=0

Now solve for w for each factor:

w=2 or w=-2


So our possible solutions are:

w=2 or w=-2

However, if we plug in w=-2 into the original equation, we'll get a zero denominator. So w=-2 is not a possible solution



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Answer:


So our solution is w=2