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Question 1209290: Solve the inequality x(x + 6) > 16 - x^2 + 14x. Write your answer in interval notation.
Found 2 solutions by mccravyedwin, math_tutor2020:
Answer by mccravyedwin(408)   (Show Source):
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Consider the equation 
Let's get everything to one side.
That last equation is of the format 
where,
a = 1, b = -4, c = -8
Plug those into the quadratic formula.
 or 
 or  both of which are approximate.
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Draw a number line with regions A, B, C
region A = stuff to the left of -1.464
region B = stuff between -1.464 and 5.464
region C = stuff to the right of 5.464
Possible test values for regions A,B,C could be: x = -2, x = 0, x = 6 in that order
You plug each test value back into the original inequality  to see if we get a true statement or not.
If we tried x = -2, then,
which is true.
Any x value in region A is in the solution set for
Region A is  aka 
Yielding the interval notation 
Use curved parenthesis to indicate we do not include the endpoints in the solution set.
If we tried x = 0 then
which is false.
Any x value in region B, ie the region between  and  will make the original inequality false.
The last region to test is region C.
Let's plug in x = 6
which is true
Therefore  aka  translates to the interval notation 
We found these two interval notation regions
or
Glue them together with a set union symbol to end up with this final answer
You can use a graphing calculator like Desmos and GeoGebra to confirm.
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