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
To solve the inequality (3x + 11)(x - 1) ≥ 0, we first find the critical points where the expression equals zero. Step 1: Find the critical points by setting each factor equal to zero. Set the first factor to zero: 3x + 11 = 0 3x = -11 x = -(11)/(3) Set the second factor to zero: x - 1 = 0 x = 1 The critical points are x = -(11)/(3) and x = 1. Step 2: Use the critical points to divide the number line into intervals. The critical points are x = -(11)/(3) ≈ -3.67 and x = 1. These points divide the number line into three intervals: 1. x ≤ -(11)/(3) 2. -(11)/(3) ≤ x ≤ 1 3. x ≥ 1 Step 3: Test a value from each interval in the original inequality (3x + 11)(x - 1) ≥ 0. Alternatively, we can observe the shape of the quadratic function. When expanded, (3x + 11)(x - 1) = 3x^2 + 8x - 11. Since the coefficient of x^2 is positive (3 > 0), the parabola opens upwards. This means the expression is greater than or equal to zero outside or at its roots. So, the inequality (3x + 11)(x - 1) ≥ 0 is satisfied when x is less than or equal to the smaller root, or greater than or equal to the larger root. This means x ≤ -(11)/(3) or x ≥ 1. Step 4: Write the solution set. The solution set in interval notation is (-, -(11)/(3)] [1, ). The final answer is (-, -(11)/(3)] [1, ). 3 done, 2 left today. You're making progress.