If 4 ≤ x ≤ 8 and 2 ≤ y ≤ 7, then what is the ratio of maximum value of (x + y) to minimum value of (x − y)?
None of the above
Understanding the Problem with Inequalities
The question asks us to find the ratio of the maximum possible value of the sum of two variables, $\left(x + y\right)$, to the minimum possible value of the difference of the same two variables, $\left(x - y\right)$. We are given the ranges within which $x$ and $y$ lie using inequalities:
- For $x$: $4 \le x \le 8$. This means $x$ can be any number from 4 up to 8, including 4 and 8.
- For $y$: $2 \le y \le 7$. This means $y$ can be any number from 2 up to 7, including 2 and 7.
We need to use these ranges to find the required maximum and minimum values.
Calculating the Maximum Value of $\left(x + y\right)$
To find the maximum value of a sum like $\left(x + y\right)$, we should take the largest possible value for each variable and add them together. The largest value for $x$ in its given range is 8, and the largest value for $y$ in its given range is 7.
Maximum value of $x = 8$
Maximum value of $y = 7$
Therefore, the maximum value of $\left(x + y\right)$ is:
$\text{Max}(x + y) = \text{Max}(x) + \text{Max}(y) = 8 + 7 = 15$.
Calculating the Minimum Value of $\left(x - y\right)$
To find the minimum value of a difference like $\left(x - y\right)$, we need to make the result as small as possible. We achieve this by taking the smallest possible value for $x$ (the first term) and subtracting the largest possible value for $y$ (the second term). The smallest value for $x$ in its given range is 4, and the largest value for $y$ in its given range is 7.
Minimum value of $x = 4$
Maximum value of $y = 7$
Therefore, the minimum value of $\left(x - y\right)$ is:
$\text{Min}(x - y) = \text{Min}(x) - \text{Max}(y) = 4 - 7 = -3$.
Calculating the Ratio of Maximum $\left(x + y\right)$ to Minimum $\left(x - y\right)$
Now that we have the maximum value of $\left(x + y\right)$ and the minimum value of $\left(x - y\right)$, we can calculate their ratio.
Ratio = $\frac{\text{Maximum value of } (x + y)}{\text{Minimum value of } (x - y)}$
Ratio = $\frac{15}{-3}$
Ratio = $-5$
Comparing with the Given Options
Let's look at the options provided:
- Option 1: 6
- Option 2: 15/2 = 7.5
- Option 3: -15/2 = -7.5
- Option 4: None of the above
Our calculated ratio is -5, which is not equal to 6, 7.5, or -7.5. Therefore, the correct answer is "None of the above".
Revision Table: Key Values
| Expression | Calculation | Value |
|---|---|---|
| Maximum $\left(x + y\right)$ | Max($x$) + Max($y$) = 8 + 7 | 15 |
| Minimum $\left(x - y\right)$ | Min($x$) - Max($y$) = 4 - 7 | -3 |
| Ratio | $\frac{\text{Max}(x+y)}{\text{Min}(x-y)} = \frac{15}{-3}$ | -5 |
Additional Information on Range and Extreme Values
When dealing with ranges of variables defined by inequalities, finding the maximum or minimum values of expressions involving these variables requires careful consideration of how the operations (addition, subtraction, multiplication, division) affect the bounds.
- Sum ($\boldsymbol{a + b}$): To find the maximum value of $\left(a + b\right)$, add the maximum values of $a$ and $b$. To find the minimum value of $\left(a + b\right)$, add the minimum values of $a$ and $b$.
- Difference ($\boldsymbol{a - b}$): To find the maximum value of $\left(a - b\right)$, take the maximum value of $a$ and subtract the minimum value of $b$. To find the minimum value of $\left(a - b\right)$, take the minimum value of $a$ and subtract the maximum value of $b$.
- Product ($\boldsymbol{a \times b}$): Finding the maximum/minimum of $\left(a \times b\right)$ depends on the signs of the ranges of $a$ and $b$. You need to check the products of the extreme values (min $\times$ min, min $\times$ max, max $\times$ min, max $\times$ max) to find the true maximum and minimum.
- Quotient ($\boldsymbol{a / b}$): Similar to product, finding the maximum/minimum of $\left(a / b\right)$ is more complex and depends on the signs and values of the extremes, especially ensuring the denominator's range does not include zero.
In this problem, we focused on sum and difference, applying the rules for these operations on variable ranges.
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