The secret to being able to do quadratic inequalities is in visualising the graph of the quadratic. Once we have that in our minds (or on the page --> NOT on a calcualtor!), then we can instantly see the answer.
The crux of sketching any graph is in knowing the roots... those points where the curve crosses the x-axis. This is the standard problem of solving any quadratic.
Inequalities merely ask you to quote the values of x where the graph is either:
above the x-axis (where y>0), or
below the x-axis (where y<0)
, to find the roots, i.e. where the curve crosses the x-axis.
answer for this is at x=-1 and x=-2, so you should then be imagining the graph as
Everything so far is "normal" quadratic equations. Now we do the inequality part, once we have the sketch of the curve in our heads.
The question wants to know "which x-values do we need that will give a quadratic function value less than zero".
--> This is all x-values between x=-2 and x=-1 --> we write this as -2 < x < -1.
Again, solve the "equal" version first, to find the roots...those points where the curve crosses the x-axis. Sketch it and then do the "inequality" part: The answer would be the 2 outer regions.
--> hence x< -2 and x>-1
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