Solve the quadratic inequality: \(x^2 - 4x - 5 \leq 0\).
Answer
Step 6: The test results in the inequality being false for \(-2\) and \(6\), but true for \(0\). Therefore, the solution to the inequality is \(-1 \leq x \leq 5\).
Step 1 ：Step 1: First, we rewrite the inequality as an equation to find critical points. The equation is, \(x^2 - 4x - 5 = 0\).
Step 2 ：Step 2: Factor the equation. The factored form of the equation is \((x - 5)(x + 1) = 0\).
Step 3 ：Step 3: Solve for \(x\). This gives us two solutions: \(x = 5\) and \(x = -1\).
Step 4 ：Step 4: Next, we use these solutions to separate the number line into three intervals: \(-\infty, -1\), \(-1, 5\), and \(5, \infty\).
Step 5 ：Step 5: We take a test point from each interval and substitute it into the original inequality. For \(-\infty, -1\), we can pick \(-2\); for \(-1, 5\), we can pick \(0\); for \(5, \infty\), we can pick \(6\).
Step 6 ：Step 6: The test results in the inequality being false for \(-2\) and \(6\), but true for \(0\). Therefore, the solution to the inequality is \(-1 \leq x \leq 5\).
