the inequality $x^2 + 12x + 35 \ge 0$ has two critical points and three possible intervals for solutions. choose each set of possible test points for the three intervals.

-8, -6, -4
-10, -6, 0
-6, 0, 6
-6, 0, 10

Question

the inequality $x^2 + 12x + 35 \ge 0$ has two critical points and three possible intervals for solutions. choose each set of possible test points for the three intervals.

-8, -6, -4
-10, -6, 0
-6, 0, 6
-6, 0, 10

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Quadratic Inequalities" ], "new_concepts": [], "current_concepts": [ "Quadratic Inequalities", "Zeros of Quadratic Functions" ] } ## Find the critical points of the quadratic expression $$ \begin{aligned} &x^2 + 12x + 35 = 0 \ &(x + 7)(x + 5) = 0 \ &x = -7, \quad x = -5 \end{aligned} $$ ## Determine the three intervals $$ \begin{aligned} &\text{Interval 1: } (-\infty, -7) \ &\text{Interval 2: } (-7, -5) \ &\text{Interval 3: } (-5, \infty) \end{aligned} $$ ## Identify the set containing one test point from each interval $$ \begin{aligned} &\text{For } \{-10, -6, 0\}: \ &-10 \in (-\infty, -7) \ &-6 \in (-7, -5) \ &0 \in (-5, \infty) \end{aligned} $$

-8, -6, -4

-10, -6, 0

-6, 0, 6

-6, 0, 10

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Inequalities" ] }

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QUESTION IMAGE

Question

Explanation:

Find the critical points of the quadratic expression

Determine the three intervals

Identify the set containing one test point from each interval

Answer: