-2x (x+4) ≥ 6x +16

To solve the inequality -2x(x+4) ≥ 6x + 16, we can follow these steps: Step 1: Simplify the expression on both sides of the inequality. -2x(x+4) ≥ 6x + 16 -2x^2 - 8x ≥ 6x + 16 Step 2: Move all the terms to one side of the inequality to set it equal to zero. -2x^2 - 8x - 6x - 16 ≥ 0 -2x^2 - 14x - 16 ≥ 0 Step 3: Solve the quadratic equation -2x^2 - 14x - 16 = 0 by factoring or using the quadratic formula. Factoring: -2x^2 - 14x - 16 = 0 (-2x - 2)(x + 8) = 0 Setting each factor equal to zero: -2x - 2 = 0 or x + 8 = 0 -2x = 2 or x = -8 x = -1 or x = -8 Step 4: Plot the solutions on a number line and determine the sign of the quadratic expression (-2x^2 - 14x - 16) in each interval. Number Line: -----------------o--------------o----------------- -8 -1 In the interval (-∞, -8), the quadratic expression is positive. In the interval (-8, -1), the quadratic expression is negative. In the interval (-1, +∞), the quadratic expression is positive. Step 5: Determine the solution to the inequality based on the sign of the quadratic expression in each interval. Since we want to solve the inequality -2x(x+4) ≥ 6x + 16, we are looking for values of x that make the left side greater than or equal to the right side. From the number line, we see that the quadratic expression is positive in the intervals (-∞, -8) and (-1, +∞). Therefore, the solution to the inequality is: x ≤ -8 or x > -1 Note: The inequality is inclusive for x ≤ -8 because the original inequality includes "greater than or equal to" (≥).