SOLUTION: What is the solution set to {{{(x-4)/(x+3)>=2}}}

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Question 140619: What is the solution set to %28x-4%29%2F%28x%2B3%29%3E=2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-4%29%2F%28x%2B3%29%3E=2 Start with the given inequality


%28x-4%29%2F%28x%2B3%29-2%3E=0 Subtract 2 from both sides


%28x-4%29%2F%28x%2B3%29-%282%29%28x%2B3%29%2F%28x%2B3%29%3E=0 Multiply 2 by %28x%2B3%29%2F%28x%2B3%29


%28x-4%29%2F%28x%2B3%29-%282%28x%2B3%29%29%2F%28x%2B3%29%3E=0 Multiply


%28x-4-2%28x%2B3%29%29%2F%28x%2B3%29%3E=0 Combine the fractions


%28x-4-2x-6%29%2F%28x%2B3%29%3E=0 Distribute


%28-x-10%29%2F%28x%2B3%29%3E=0 Combine like terms


x%2B3=0 Set the denominator equal to zero


x=0-3Subtract 3 from both sides


x=-3 Combine like terms on the right side


So the vertical asymptote is x=-3. So one critical value is x=-3

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-x-10=0 Set the numerator equal to zero


-10=x Add x to both sides


So another critical value is x=-10




So our critical values are x=-3 and x=-10

Now set up a number line and plot the critical values on the number line

number_line%28+600%2C+-15%2C+10%2C-3%2C-10%29



So let's pick some test points that are near the critical values and evaluate them.


Let's pick a test value that is less than -10 (notice how it's to the left of the leftmost endpoint):

So let's pick x=-11

%28x-4%29%2F%28x%2B3%29%3E=2 Start with the given inequality


%28-11-4%29%2F%28-11%2B3%29%3E=+2 Plug in x=-11


1.875%3E=2 Evaluate and simplify the left side




Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.




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Let's pick a test value that is in between -10 and -3:

So let's pick x=-6

%28x-4%29%2F%28x%2B3%29%3E=2 Start with the given inequality


%28-6-4%29%2F%28-6%2B3%29%3E=+2 Plug in x=-6


3.3333%3E=+2 Evaluate and simplify the left side



Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.

So part our solution in interval notation is [)



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Let's pick a test value that is greater than -3 (notice how it's to the right of the rightmost endpoint):

So let's pick x=-2

%28x-4%29%2F%28x%2B3%29%3E=2 Start with the given inequality


%28-2-4%29%2F%28-2%2B3%29%3E=+2 Plug in x=-2


-6%3E=+2 Evaluate and simplify the left side

Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.




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Summary:

So the solution interval notation is:

[)