Question 1141444: How would I solve for x on this quadratic inequality? this is the one that's been puzzling me: (x+5)(x-2)(x-7)≤0
Here are the multiple choice answers: https://i.imgur.com/oEmAOtH.png
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20849)   (Show Source):
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
In the solution by tutor @mathlover1, she identified the values of x that make each of the individual factors negative or zero; but it is unclear how she used that information to identify the intervals on which the complete expression is negative or zero.
Consider the polynomial function
We want to identify when the function value is zero or negative.
Clearly the function value is zero at x=-5, x=2, and x=7.
To identify the intervals on which the function value is negative, there are several options. Among those options are
(1) The function is a cubic with positive leading coefficient. Think of what the graph of that kind of function looks like. The function value is negative for "large negative" value of x and positive for large positive values of x. That, along with the known zeros of the function, will determine the intervals on which the function value is negative.
(2) Choose a test point in each of the intervals determined by the zeros of the function to see whether the function value is positive or negative in each interval.
(3) (My personal choice....) You know that for large negative values of x all three factors are negative, so the function value is negative to the left of x=-5. Then, as you "walk" along the number line to the right, the function value changes sign each time you pass one of the zeros. So the function value is (a) negative for x less than -5; (b) positive for x between -5 and 2; (c) negative for x between 2 and 7; and (d) positive for x greater than 7.
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