Question

solve the quadratic inequality.
\\((x + 5)(x + 2) > 0\\)

select the correct choice below and fill in the answer boxes within your choice.
\\(\bigcirc\\) a. the solutions are \\(\square < x < \square\\).
\\(\bigcirc\\) b. the solutions are \\(x < \square\\) or \\(x > \square\\).

Explanation:

Step1: Find critical points

Set each factor equal to zero: \(x + 5 = 0\) gives \(x = -5\); \(x + 2 = 0\) gives \(x = -2\). These are the critical points that divide the number line into intervals.

Step2: Test intervals

• For \(x < -5\) (e.g., \(x = -6\)): \((-6 + 5)(-6 + 2)=(-1)(-4)=4>0\), so this interval is part of the solution.
• For \(-5 < x < -2\) (e.g., \(x = -3\)): \((-3 + 5)(-3 + 2)=(2)(-1)= -2<0\), not part of the solution.
• For \(x > -2\) (e.g., \(x = 0\)): \((0 + 5)(0 + 2)=(5)(2)=10>0\), so this interval is part of the solution.

Step3: Determine solution type

Since the product is positive when \(x < -5\) or \(x > -2\), we choose option B.

Answer:

B. The solutions are \(x < -5\) or \(x > -2\).