+23246 To solve the inequality: x3 + 4x > 5x2
x3 - 5x2 + 4x > 0
x(x2 - 5x + 4) > 0
x(x -1)(x - 4) > 0
The points that divide the number line into regions are 0, 1, and 4.
[None of these points are possible answers because the problem has > not >=.]
We need to check one point from each of these regions: x < 0, 0 < x < 1, 1 < x < 4, x > 4
Region: x < 0: We can use any point, I will choose -5 and place this number into x(x -1)(x - 4) > 0.
-5 makes x negative; -5 makes (x - 1) negative; -5 makes (x - 4) negative; multiplying three negatives together,
we get a negative number, not the positive that the problem says. This region doesn't work.
Region: 0 < x < 1: We can use any point; I will choose ½.
½ makes x positive; ½ makes (x - 1) negative; ½ makes (x - 4) negative; multiplying a positive times a negative
times another negative gives us a positive number, just what we need. This region works.
Region: 1 < x < 4: We can use any point, I will choose 2.
2 makes x positive; 2 makes (x - 1) positive; 2 makes (x - 4) negative; multiplying a positive times a positive
times a negative gives us a negative number. This region doesn't work.
Region: x > 4: We an use any point, I will choose 10.
10 makes x positive; 10 makes (x - 1) positive; 10 makes (x - 4) positive; multiply all positive numbers together
gives us a positive number. This region works.
The answer consists of the regions that work. 0 < x < 1 and x > 4.
