To determine the values of x that satisfy x² - 7x + 10 ≤ 0, we need to find the roots of the quadratic equation x² - 7x + 10 = 0 and then analyze the behavior of the quadratic function.
To find the roots, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
In this case, a = 1, b = -7, and c = 10, so:
x = (-(-7) ± √((-7)² - 4(1)(10))) / 2(1)
x = (7 ± √9) / 2
x1 = 5 and x2 = 2
The roots of the equation are x = 5 and x = 2.
Now, we need to analyze the behavior of the quadratic function in the intervals between the roots and to the left and right of the roots. We can do this by creating a sign chart:
Interval | x² - 7x + 10
---------------------------------
(-∞, 2) | +
(2, 5) | -
(5, +∞) | +
In the interval (-∞, 2), the quadratic function is positive because all the factors are positive. In the interval (2, 5), the function is negative because x - 2 is negative and x - 5 is positive. In the interval (5, +∞), the function is positive again because both factors are positive.
Therefore, the solution to the inequality x² - 7x + 10 ≤ 0 is the interval [2, 5], because the function is non-positive in that interval and positive elsewhere. In interval notation, we can write:
2 ≤ x ≤ 5
So the correct answer is:
2 ≤ x ≤ 5.