SOLUTION: Solve the inequality {{{(x-4)/(x-7)>=0}}}
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Question 149240: Solve the inequality
Answer by jim_thompson5910(35256) (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
Set the denominator equal to zero
Add 7 to both sides
Combine like terms on the right side
So the vertical asymptote is
--------------------
Now we need to find any x-intercepts
Start with the given equation
Plug in
Multiply both sides by .
Add 4 to both sides.
So the x-intercept is (4,0)
---------------
This means that we'll have to test three regions
Region 1:
This region is from negative infinity to the x-intercept
So let's test the value
Start with the given equation
Plug in
Simplify.
Since 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
-----------
Region 2:
This region is from the x-intercept to the vertical asymptote
So let's test the value
Start with the given equation
Plug in
Simplify.
Since 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
-----------
Region 3:
This region is from the vertical asymptote to positive infinity
So let's test the value
Start with the given equation
Plug in
Simplify.
Since 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
So that means that the solution is
(]
()
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