for this inequality:
(x-1)(x-2)(x-3) <0 becomes
x<1 x<2 x<3
but why when I graph the inequality it becomes (this is the actually answer) {x|x<1 or 2<x<3}
I don't understand why it couldn't just be x<3 ?
Also I don't understand why there's an "or" and why the < changed?
in the original form, the PRODUCT is less than zero, which means either one or three of the terms is negative.
so, if x<1 works (all three are negative), as does 2<x<3 (one term is negative)
How would I know its 2<x<3?
the steps to getting x<1 and 2<x<3
When you plotted the graph of the function, did you notice that the graph crosses the x-axis at x=1, x=2 and x=3?
The graph stays below the x-axis when x<1, and also when 2<x<3.
x<1 when all three factors are negative, and 2<x<3 when only one term (x-3) is negative, as Mr. Pursley mentioned.
To understand why the solution to the inequality (x-1)(x-2)(x-3) < 0 is {x|x<1 or 2<x<3}, let's break it down step by step:
1. Start by finding the critical points, where the expression (x-1)(x-2)(x-3) becomes zero. In this case, the critical points are x=1, x=2, and x=3.
2. Now, let's analyze each interval between the critical points. We divide the number line into four intervals: (-∞, 1), (1, 2), (2, 3), and (3, +∞).
3. Next, we choose a test point from each interval, and substitute it into the expression (x-1)(x-2)(x-3) to determine whether the expression is positive or negative in that interval.
- For the interval (-∞, 1), you can choose a test point such as x = 0. Substituting x=0 into the expression results in a negative value, indicating that the expression (x-1)(x-2)(x-3) is negative in this interval.
- For the interval (1, 2), you can choose x = 1.5 as a test point. Substituting x=1.5 into the expression yields a positive value, indicating that the expression (x-1)(x-2)(x-3) is positive in this interval.
- For the interval (2, 3), you can choose x = 2.5 as a test point. Substituting x=2.5 into the expression gives a negative value, indicating that the expression (x-1)(x-2)(x-3) is negative in this interval.
- Finally, for the interval (3, +∞), you can choose x = 4 as a test point. Substituting x=4 into the expression results in a positive value, indicating that the expression (x-1)(x-2)(x-3) is positive in this interval.
4. Now, we consider the signs of the expression in the intervals. If the expression (x-1)(x-2)(x-3) is negative, it means that it is less than zero (0). If it is positive, it is greater than zero (0).
- From our analysis, we find that the expression is negative in the intervals (-∞, 1) and (2, 3).
5. Combining the intervals where the expression (x-1)(x-2)(x-3) is negative, we get the solution as x < 1 or 2 < x < 3.
Thus, the correct solution to the inequality (x-1)(x-2)(x-3) < 0 is {x|x<1 or 2<x<3}. The "or" in the solution indicates that either condition (x<1 or 2<x<3) can be satisfied for the inequality to hold true.
Regarding the change in the inequality symbol, when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. In this case, when (x-1)(x-2)(x-3) is negative, it means it is less than zero (0), so we use the inequality symbol "<" instead of ">". This is why the inequality symbol changes for different intervals.