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: Rewrite the inequality by moving all terms to one side. (x^2-12)/(x) > 1 (x^2-12)/(x) - 1 > 0 (x^2-12-x)/(x) > 0 (x^2-x-12)/(x) > 0 Step 2: Factorize the numerator. The quadratic expression x^2-x-12 can be factored as (x-4)(x+3). So the inequality becomes: ((x-4)(x+3))/(x) > 0 Step 3: Identify the critical points. The critical points are the values of x that make the numerator or the denominator equal to zero. From the numerator: x-4=0 x=4 and x+3=0 x=-3. From the denominator: x=0. The critical points are x=-3, 0, 4. Step 4: Test intervals on the number line defined by the critical points. The critical points divide the number line into four intervals: (-, -3), (-3, 0), (0, 4), and (4, ). For x < -3 (e.g., x=-4): (x-4) is negative. (x+3) is negative. x is negative. The expression is ((-)(-))/((-)) = ((+))/((-)) = (-). So, ((x-4)(x+3))/(x) < 0. For -3 < x < 0 (e.g., x=-1): (x-4) is negative. (x+3) is positive. x is negative. The expression is ((-)(+))/((-)) = ((-))/((-)) = (+). So, ((x-4)(x+3))/(x) > 0. This interval is part of the solution. For 0 < x < 4 (e.g., x=1): (x-4) is negative. (x+3) is positive. x is positive. The expression is ((-)(+))/((+)) = ((-))/((+)) = (-). So, ((x-4)(x+3))/(x) < 0. For x > 4 (e.g., x=5): (x-4) is positive. (x+3) is positive. x is positive. The expression is ((+)(+))/((+)) = ((+))/((+)) = (+). So, ((x-4)(x+3))/(x) > 0. This interval is part of the solution. Step 5: Combine the intervals where the inequality is satisfied. The inequality ((x-4)(x+3))/(x) > 0 holds for -3 < x < 0 or x > 4. The range of values for x is (-3, 0) (4, ).