Find the number of integer solutions x for which x^2 - 7x + 10 <= 0.
Correct: E
We need to find the integer solutions for the inequality x^2 - 7x + 10 <= 0.
First, factor the quadratic expression:
x^2 - 7x + 10 = (x-2)(x-5).
So the inequality becomes (x-2)(x-5) <= 0.
The roots of the quadratic equation x^2 - 7x + 10 = 0 are x = 2 and x = 5.
Since the quadratic is a parabola opening upwards (coefficient of x^2 is positive), its value is less than or equal to zero between its roots.
Therefore, the inequality (x-2)(x-5) <= 0 is satisfied when 2 <= x <= 5.
We are looking for integer solutions within this range. The integers are:
x = 2
x = 3
x = 4
x = 5
There are 4 integer solutions.