Question

what are the solutions to the inequality (x-3)(x+5)≤0?
○ {x| 3≤x≤5}
○ {x| -5≤x≤-3}
○ {x| -5≤x≤3}
○ {x| -3≤x≤5}

Explanation:

Step1: Find critical points

Set each factor equal to zero: \(x - 3 = 0\) gives \(x = 3\); \(x + 5 = 0\) gives \(x = -5\). These are the critical points that divide the number line into intervals.

Step2: Test intervals

• For \(x < -5\) (e.g., \(x = -6\)): \((-6 - 3)(-6 + 5)=(-9)(-1)=9>0\), so not a solution.
• For \(-5 < x < 3\) (e.g., \(x = 0\)): \((0 - 3)(0 + 5)=(-3)(5)=-15\leq0\), so this interval is a solution.
• For \(x > 3\) (e.g., \(x = 4\)): \((4 - 3)(4 + 5)=(1)(9)=9>0\), so not a solution.

Step3: Include critical points

Since the inequality is \(\leq0\), we include \(x = -5\) and \(x = 3\) (where the product is zero). So the solution is \(-5\leq x\leq3\).

Answer:

\(\{x|-5\leq x\leq3\}\)