This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Solve the corresponding equation x^2 - 5x - 6 = 0 to find the boundary points. Apply the quadratic formula: x = -b ± sqrt(b^2 - 4ac)2a where a = 1, b = -5, c = -6. Compute the discriminant: b^2 - 4ac = (-5)^2 - 4(1)(-6) = 25 + 24 = 49 Substitute: x = 5 ± sqrt(49)2 = (5 ± 7)/(2) x = (5 + 7)/(2) = (12)/(2) = 6, x = (5 - 7)/(2) = (-2)/(2) = -1 The roots are x = -1 and x = 6. Step 2: Factor the quadratic: x^2 - 5x - 6 = (x + 1)(x - 6) Step 3: Determine the sign of (x + 1)(x - 6) in the test intervals (-, -1), (-1, 6), and (6, ). For x = -2 in (-, -1): (-2 + 1)(-2 - 6) = (-1)(-8) = 8 > 0 For x = 0 in (-1, 6): (0 + 1)(0 - 6) = (1)(-6) = -6 < 0 For x = 7 in (6, ): (7 + 1)(7 - 6) = (8)(1) = 8 > 0 The expression is negative or zero between the roots, and the parabola opens upwards (since a > 0). Thus, x^2 - 5x - 6 ≤ 0 for -1 ≤ x ≤ 6. Step 4: Express in interval notation: [-1, 6]. Final answer: [-1,6]