goodyear2013 wrote:
Which of the following describes all the values of x for which x(x−1)(x−2)≥0?
A. 0 ≤ x ≤ 1 or x ≥ 2
B. x ≤ 0 or x ≥ 2
C. 0 ≤ x ≤ 1
D. 1 ≤ x ≤ 2 or x ≥ 2
E. 0 ≤ x ≤ 2
x(x−1)(x−2)≥0 --> the "roots" are 0, 1 and 2, this gives us 4 ranges:
x<0;
0≤x≤1;
1<x<2;
x≥2.
Next, test some extreme value for x: if x is some large enough number, say 10, then all three multiples will be positive which gives the positive result for the whole expression, so when x≥2, the expression is positive. Now the trick: as in the 4th range expression is positive, then in the 3rd it'll be negative, in the 2nd it'll ne positive and finally in the 1st it'll be negative: - + - + . So, the ranges when the expression is positive or 0 are: 0≤x≤1 and x≥2.
Answer: A.
The graph inequalities strategy appears good for 2 solutions, can you kindly explain how you are assigning signs here and if the approach is always applicable.