Solve the inequality.

Question

(x-3)(x+1)(x+4)≤0

Answer 1

Use the Dot Cross technique, explained below.

 

THE DOT-CROSS TECHNIQUE
Consider a polynomial function of degree three: (x+1)(x-1)(2x-5) = f(x)

Analysis of functions by the Dot-Cross technique requires that, first, all parts be fully factored.
The zeros of this cubic polynomial are -1, 1 and 5/2. We place them in order on a number-line:
---------------------x-----------------------x------------------------x--------------------------
                         -1                             1                            2.5

The x symbol is placed at points -1, 1 and 2.5 on the number-line because the associated factor was of odd degree (odd multiplicity, namely one) in each case. If multiplicity had been even, the DOT symbol would have been used, instead.

The behavior of any polynomial is such that, on an interval between any two zeros, function values all have the same sign. We observe that, for Therefore we know that the function is always positive, or above the x-axis, on the rightmost interval from 2.5 to infinity.

 
Now it is immediately possible to determine the behavior on every other interval, because, as moves leftward through the x symbol at 2.5 (to get into the interval [1, 2.5], the function changes sign; that is, the behavior changes whenever transits through a cross (x). On the other hand, whenever transits through a dot (• ), the function values don’t change sign.

From these facts, the graph can be sketched meaningfully.