I notice that the sum consists of three absolute value expressions, each of which is linear in $x$. This means that each expression is either constant or has a slope of $\pm 1$, depending on the value of $x$. I also notice that the expressions inside the absolute value signs have different signs, depending on the value of $x$. For example, if $x < 1$, then $x - 1$ is negative, so $|x - 1| = -(x - 1) = 1 - x$. Similarly, if $x > 2$, then $x - 2$ is positive, so $|x - 2| = x - 2$. This means that the sum will change its slope at the points where the expressions inside the absolute value signs change their signs, which are $x = 1$, $x = 1.5$, and $x = 2$. These are the critical points of the sum, and I can use them to divide the number line into four intervals: $(-\infty, 1)$, $(1, 1.5)$, $(1.5, 2)$, and $(2, \infty)$. I can then find the slope of the sum in each interval by adding the slopes of the absolute value expressions. For example, in the interval $(1, 1.5)$, the slopes of the absolute value expressions are $1$, $-1$, and $-1$, so the slope of the sum is $1 - 1 - 1 = -1$. This means that the sum is decreasing in this interval, so its maximum value occurs at the left endpoint, $x = 1$, and its minimum value occurs at the right endpoint, $x = 1.5$. I can repeat this process for the other intervals and find that the sum has a slope of $-3$ in $(-\infty, 1)$, a slope of $1$ in $(1.5, 2)$, and a slope of $3$ in $(2, \infty)$. This means that the sum is decreasing in $(-\infty, 1)$ and $(1, 1.5)$, and increasing in $(1.5, 2)$ and $(2, \infty)$. Therefore, the minimum value of the sum must occur at one of the critical points, where the slope changes from negative to positive. To find the minimum value, I just need to plug in the values of the critical points into the sum and compare them. At $x = 1$, the sum is $|1 - 1| + |1 - 1.5| + |1 - 2| = 0 + 0.5 + 1 = 1.5$. At $x = 1.5$, the sum is $|1.5 - 1| + |1.5 - 1.5| + |1.5 - 2| = 0.5 + 0 + 0.5 = 1$. At $x = 2$, the sum is $|2 - 1| + |2 - 1.5| + |2 - 2| = 1 + 0.5 + 0 = 1.5$. Therefore, the minimum value of the sum is 1, which occurs at $x = 1.5$. 
What is the least possible value of the sum $|x - 1| + |x - 1.5| + |x - 2|$?