SOLUTION: Solve the inequality {{{(x-4)/(x-7)>=0}}}

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Question 149240: Solve the inequality %28x-4%29%2F%28x-7%29%3E=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, we need to find the vertical asymptote(s)

Vertical Asymptote:
To find the vertical asymptote, just set the denominator equal to zero and solve for x

x-7=0 Set the denominator equal to zero


x=0%2B7Add 7 to both sides


x=7 Combine like terms on the right side


So the vertical asymptote is x=7

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Now we need to find any x-intercepts


y=%28x-4%29%2F%28x-7%29 Start with the given equation


0=%28x-4%29%2F%28x-7%29 Plug in y=0


0=x-4 Multiply both sides by x-7.


4=x Add 4 to both sides.


So the x-intercept is (4,0)


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This means that we'll have to test three regions


Region 1:

This region is from negative infinity to the x-intercept x=4

So let's test the value x=0

y=%28x-4%29%2F%28x-7%29 Start with the given equation


y=%280-4%29%2F%280-7%29 Plug in x=0


y=4%2F7 Simplify.

Since 4%2F7 is greater than or equal to zero, this means that every point in the interval (] is above the x-axis.

So the interval (] is part of the solution to the inequality %28x-4%29%2F%28x-7%29%3E=0

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Region 2:

This region is from the x-intercept x=4 to the vertical asymptote x=7

So let's test the value x=5

y=%28x-4%29%2F%28x-7%29 Start with the given equation


y=%285-4%29%2F%285-7%29 Plug in x=5


y=-1%2F2 Simplify.

Since -1%2F2 is not greater than or equal to zero, this means that every point in the interval () is below the x-axis.

So the interval () is not part of the solution to the inequality %28x-4%29%2F%28x-7%29%3E=0



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Region 3:

This region is from the vertical asymptote x=7 to positive infinity

So let's test the value x=8

y=%28x-4%29%2F%28x-7%29 Start with the given equation


y=%288-4%29%2F%288-7%29 Plug in x=8


y=4 Simplify.

Since 4 is greater than or equal to zero, this means that every point in the interval () is above the x-axis.

So the interval () is part of the solution to the inequality %28x-4%29%2F%28x-7%29%3E=0


So that means that the solution is

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