Final answer:
The possible values of the number can be found by solving the quadratic inequality x * (x - 4) ≤ 64 and finding the intervals that satisfy the inequality.
Step-by-step explanation:
To find the possible values of the number, let's set up the equation based on the given information. We have the product of x and the sum of a number and -4 is less than or equal to 64. In mathematical terms, this can be represented as x * (x + (-4)) ≤ 64. Simplifying this further, we have x * (x - 4) ≤ 64.
Next, we can solve this quadratic inequality. To do that, we need to find the critical points. The critical points occur when x * (x - 4) = 64. Setting the equation to 0, we get x² - 4x - 64 = 0. Solving this quadratic equation, we find the critical points to be x = 8 and x = -4.
Now, we have three intervals on the number line: (-∞, -4), (-4, 8), and (8, +∞). We need to test a value in each interval to determine whether it satisfies the inequality. If it does, then that interval is part of the solution.
For example, let's test a value in the first interval such as x = -5. Plugging it into the original inequality, we get (-5) * ((-5) - 4) ≤ 64, which simplifies to -5 * (-9) ≤ 64. This condition is true, meaning the first interval is part of the solution set. Similarly, you can test values in the other intervals to determine which values satisfy the inequality.