Question

the inequality $x^2 + 12x + 35 \geq 0$ has two critical points and three possible intervals for solutions. choose each set of possible test points for the three intervals.
$\square$ $-8, -6, -4$
$\square$ $-10, -6, 0$
$\square$ $-6, 0, 6$
$\square$ $-6, 0, 10$
done

Explanation:

Step1: Find critical points

First, solve the equation \(x^{2}+12x + 35=0\). Factor the quadratic: \(x^{2}+12x + 35=(x + 5)(x+7)=0\). So the critical points are \(x=-7\) and \(x=-5\).

Step2: Determine intervals

The intervals are \((-\infty,-7)\), \((-7, - 5)\), and \((-5,\infty)\). We need one test point from each interval.

• For \((-\infty,-7)\), a test point could be \(-10\) (since \(-10<-7\)).
• For \((-7, - 5)\), a test point could be \(-6\) (since \(-7<-6<-5\)).
• For \((-5,\infty)\), a test point could be \(0\) (since \(0>-5\)).

So the set \(-10,-6,0\) has one test point from each of the three intervals.

Answer:

\(-10, - 6, 0\) (the second option: \(-10, -6, 0\))