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The product of , and the sum of a number and - 4 is less than or equal to 64.

What are all the possible values of the number? Show your work.

User Janar
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2 Answers

24 votes
24 votes

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.

User EgorD
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3.2k points
21 votes
21 votes

Answer:

6

Step-by-step explanation:

User Ysdx
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2.8k points