Final answer:
The solution set for the inequality x²/2 > -7 -x/2 is found by rearranging it into a quadratic equation, solving it using the quadratic formula to find its critical points, and then testing intervals around these points to determine where the inequality holds true.
Step-by-step explanation:
The solution set of the inequality x²/2 > -7 -x/2 is found by solving the related quadratic equation. First, we need to rearrange the inequality to make one side equal to zero, by adding 7 and x/2 to both sides. After doing so, we have x² + x/2 - 14/2 > 0, which simplifies to x² + x - 7 > 0.
We can then solve the corresponding quadratic equation x² + x - 7 = 0 using the quadratic formula, x = (-b ± √(b² - 4ac))/(2a). In this case, a = 1, b = 1, and c = -7. Following the quadratic formula, we'll find the values of x that make the equation equal to zero, which are the critical points that separate the regions where the inequality is true from where it is false.
Finally, we test intervals defined by these critical points to find the regions where the original inequality holds true. This process will give us the solution set for the inequality in question, which represents a range or ranges of values where x satisfies the original inequality.