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

Question 140619: What is the solution set to
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given inequality

Subtract 2 from both sides

Multiply 2 by

Multiply

Combine the fractions

Distribute

Combine like terms

Set the denominator equal to zero

Subtract 3 from both sides

Combine like terms on the right side

So the vertical asymptote is . So one critical value is

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Set the numerator equal to zero

Add x to both sides

So another critical value is

So our critical values are and

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

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 (notice how it's to the left of the leftmost endpoint):

So let's pick

Start with the given inequality

Plug in

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 and :

So let's pick

Start with the given inequality

Plug in

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 (notice how it's to the right of the rightmost endpoint):

So let's pick

Start with the given inequality

Plug in

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.

---------------------------------------------------------------------------------------------

Summary:

So the solution interval notation is:

[

)

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