The value of x is to be randomly selected from the integers from 1 to 10, inclusive, and then substituted into the equation y = x^2 - 4x + 3. What is the probability that the value of y will be negative?
A. 1/10
B. 1/5
C. 1/4
D. 3/10
E. 3/5
y = (x-1)(x-3).
The CRITICAL POINTS ARE x=1 and x=3.
The critical points are the only x-values that yield y=0.
To determine the ranges where x will yield y<0, test one value to the left and one value to the right of each critical point.
Here, since x must be an integer such that 1≤x≤10, we must test only one value BETWEEN x=1 and x=3 and one value TO THE RIGHT of x=3.
1<x<3:
If we plug x=2 into y = x² - 4x + 3, we get:
y = 2² - 4*2 + 3 = -1.
Since y is negative, 1<x<3 is a valid range.
x>3:
If we plug x=4 into y = x² - 4x + 3, we get:
y = 4² - 4*4 + 3 = 3.
Since y is NOT negative, x>3 is NOT a valid range.
The only valid range is 1<x<3.
Implication:
Of the 10 integers between 1 and 10, inclusive, only one value -- x=2 -- will yield a negative value for y.
Thus:
P(y is negative) = 1/10.
The correct answer is
A.
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