Final answer:
The inequality (x² – 4)(x² – 10) < 0 is satisfied by two integers, -1 and 1, since they are the only integers between the critical points -2 and 2 where the inequality holds.
Step-by-step explanation:
To determine how many integers satisfy the inequality (x² – 4)(x² – 10) < 0, we should find the critical points where the expression changes signs. The critical points are the roots of the individual factors x² - 4 and x² - 10. The roots are x = 2, x = -2 (from x² - 4) and x = √10, x = -√10 (from x² - 10).
This gives us four intervals to examine: x < -√10, -√10 < x < -2, -2 < x < 2, and 2 < x < √10. The inequality is satisfied when the product of the two factors is less than zero, meaning we are looking for intervals where one factor is positive and the other is negative. The intervals where the inequality holds are -√10 < x < -2 and 2 < x < √10.
However, we're interested in integers, and √10 is irrational, so we only consider the integers -2, -1, 1, and 2. Since x cannot be equal to -2 or 2 due to the inequality being strictly less than zero, that leaves us with just -1 and 1 as the integers that satisfy the inequality, thus the answer is b) Two.