SOLUTION: I need help solving. Thank you x^2 - 8x <= -15

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Question 888525: I need help solving. Thank you
x^2 - 8x <= -15

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2+-+8x+%3C=+-15

Get 0 on the right, everything else on the left:

x%5E2+-+8x+%2B15+%3C=+0

%28x-3%29%28x-5%29%3C=0

The critical numbers are the numbers which either make
the left side 0 or undefined.

The critical numbers are 3 and 5.

Place these numbers on a number line:

----------------o-----o------------  
-2 -1  0  1  2  3  4  5  6  7  8  9

we test any value less than 3, say 0, in the inequality

%28x-3%29%28x-5%29%3C=0
%280-3%29%280-5%29%3C=0
%28-3%29%28-5%29%3C=0
15%3C=0

That's false so we do not shade the number line to the
left of 3

----------------o-----o------------  
-2 -1  0  1  2  3  4  5  6  7  8  9

we test any value between 3 and 5, say 4, in the inequality

%28x-3%29%28x-5%29%3C=0
%284-3%29%284-5%29%3C=0
%281%29%28-1%29%3C=0
-1%3C=0

That's true so we shade the number line between 3 and 5.

----------------o=====o------------  
-2 -1  0  1  2  3  4  5  6  7  8  9

we test any value to the right of 5, say 6, in the inequality

%28x-3%29%28x-5%29%3C=0
%286-3%29%286-5%29%3C=0
%283%29%281%29%3C=0
3%3C=0

That's false so we do not shade the number line to the right of 5.

Next we test the critical points themselves:

----------------o=====o------------  
-2 -1  0  1  2  3  4  5  6  7  8  9

Test 3

%28x-3%29%28x-5%29%3C=0
%283-3%29%283-5%29%3C=0
%280%29%28-2%29%3C=0
0%3C=0

That's true so we darken the circle at 3:

----------------●=====o------------  
-2 -1  0  1  2  3  4  5  6  7  8  9

Test 3

%28x-3%29%28x-5%29%3C=0
%285-3%29%285-5%29%3C=0
%282%29%280%29%3C=0
0%3C=0

----------------●=====●------------  
-2 -1  0  1  2  3  4  5  6  7  8  9

That's true so we darken the circle at 5 also.

The interval notation for the solution is [3,5]

The set builder notation is  matrix%281%2C5%2C++++%22%7B%22%2Cx%2C%22%7C%22%2C3%3C=x%3C=5%2C%22%7D%22+%29 

Edwin