SOLUTION: Solve (w-3)/(w+2)=(6-w)/(w-12) Please I nedd help with this! Thank you!

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Question 825884: Solve
(w-3)/(w+2)=(6-w)/(w-12)
Please I nedd help with this! Thank you!
Answer by math-vortex(648) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, there--

THE PROBLEM:
Solve for w.
(w-3)/(w+2)=(6-w)/(w-12)

A SOLUTION:

I'll go step-by-step so you can see what's happening. Multiply both sides by (w+2). This will 
clear the denominator on left side,

%28w-3%29%2F%28w%2B2%29=%286-w%29%2F%28w-12%29
%28w%2B2%29%2A%28%28w-3%29%2F%28w%2B2%29%29=%28w%2B2%29%2A%28%286-w%29%2F%28w-12%29%29
%28w-3%29=%28%28w%2B2%29%2A%286-w%29%29%2F%28w-12%29

Now multiply both sides by (w-12). This will clear denominator on the right side.
%28w-12%29%2A%28w-3%29=%28w-12%29%2A%28%28%28w%2B2%29%2A%286-w%29%29%2F%28w-12%29%29
%28w-12%29%28w-3%29=%28w%2B2%29%286-w%29

Now use the distributive property to multiply all the factors out to polynomial form.
w%5E2-12w-3w%2B36=6w-w%5E2%2B12-2w

Combine like terms and rearrange by descending degree on both sides.
w%5E2-15w%2B36=-w%5E2%2B4w%2B12

This is a quadratic equation. We need to set it equal to zero to solve. Rewrite as an equivalent 
equation with all terms on the left.
2w%5E2-19w%2B24=0

I always try to factor first because it's faster than the quadratic formula.
factors of 2w^2 are 2w and w.
factors of 24 are positive/positive or negative/negative
24 and 1, 
12 and 2, 
8 and 3,
6 and 4

We want a combination that sums to -19x when we distribute. The factors -8 and -3 will
work because (-8)(2x)+(-3(x)=-19x.

Write the quadratic equation in factored form.
%282x-3%29%28x-8%29=0

By the Zero Product Property, either

2x-3=0 OR x-8=0

Therefore, our solutions are x = 1.5 or x = 8.

Check you work by substituting these values into the original equation.

For x = 8:
%28w-3%29%2F%28w%2B2%29=%286-w%29%2F%28w-12%29
%28%280%29-3%29%2F%28%288%29%2B2%29=%286-%288%29%29%2F%28%288%29-12%29
5%2F10=%28-2%29%2F%28-4%29
1%2F2=1%2F2 CHECK!

For x=1.5:
%28w-3%29%2F%28w%2B2%29=%286-w%29%2F%28w-12%29
%28%281.5%29-3%29%2F%28%281.5%29%2B2%29=%286-%281.5%29%29%2F%28%281.5%29-12%29
%28-1.5%29%2F%283.5%29=%284.5%29%2F%28-10.5%29
-3%2F7=-3%2F7 CHECK!

Hope this helps! Feel free to email if you have any questions.

Mrs. Figgy
math.in.the.vortex@gmail.com