Final answer:
The inequality -2x² - 7x - 12 > -2x - 10 is first transformed to -2x² - 5x - 2 > 0. Critical points are found by solving the corresponding quadratic equation, followed by a sign chart to evaluate the intervals and determine the solution set.
Step-by-step explanation:
To solve the inequality -2x² - 7x - 12 > -2x - 10, we first move all terms to one side of the inequality to obtain -2x² - 5x - 2 > 0. Factoring the quadratic equation or completing the square can simplify the inequality further. Next, we need to identify the critical points by setting up the corresponding quadratic equation -2x² - 5x - 2 = 0 and solving for x, which will help us determine the intervals to test for the solution.
Once we've found the critical points, we construct a sign chart to evaluate the intervals. Multiplying or dividing by a negative number flips the direction of the inequality, as stated "Multiplying both sides of the inequality by -T reverses the sign of the inequality". After testing the intervals, we can find the solution set that satisfies the original inequality.