If -2<= x <= 2 and 3 <= y <= 8, which of the following represents the

If you think about the smallest y-x can be, that would be when y is the smallest value (3) and x is the largest (2). The minimum of y-x is 1. Same thing if you think about the largest y-x can be, which is when y is 8 and x is -2. In that case, y-x = 8 - (-2) = 10.

The answer is 1<=y-x<=10. (D) and (E) look the same as you've listed them. My answer is (D) or (E), depending which one you copied correctly.

There are other ways to do this problem that would include rules for inequalities. To me, this is the simplest, quickest way, especially if you're unsure what to do. Whenever I was in doubt, I used actual numbers instead of hypotheticals.

3 < y < 8
multiply the second one by (-1) to reverse the sign
2 > x > -2
Subtract them to get
3 - 2 < y - x < 8 - (-2)
1 < y - x < 10

possible range of (y-x) is the difference between min (y-x) and max (y-x).

min (y-x) = deduct minimum x value from max y value = 3 - 2 = 1
max (y-x) = deduct minimum x value from max y value = 8 - (-2) = 10

1 =< (y - x) <= 10

For my experience, don't use subtraction between 2 inequalities as you probably get the wrong sign

also dont use multiplication, division , esp you don't know if the variables are negative or positive or fraction

Use addition instead

-2=<x=<2
-2 <= -x <= 2
+
3<=y<=8

3 - 2 <= y - x <= 8+2

-2 <= X <= 2 ------(1)

3 <= y <= 8 --------(2)


Multiplying eq (1) by -1

2<= -x <=-2 -------(3)


Adding (1) & (3)

5<= y-x<=10

Maximum value of y-x= 8 - (-2)= 10
Minimum value of y-x= 3-2= 1

E mentions the correct range
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"Range of all possible values" for \(y - x\) is just another way of asking about the expression's minimum value and maximum value.

Minimum and maximum of \(y - x\)? Where

\(-2 \leq x \leq 2\)

\(3 \leq y \leq 8\)

To find the smallest (minimum) value for \(y - x\), you can either:

1) Keep order of subtraction in mind, and find two same-sign values that are close together (which omits -2 from calculation). Two positives for y, one for x. Closest: 3 and 2. Answer is 1. Or

2) Test all four cases

8 - 2 = 6
8 - (-2) = 10
3 - 2 = 1
3 - (-2) = 5

Minimum value of \(y - x\) = 1

Maximum?

1. Same things to keep in mind, but look for opposite-sign values that are large (or larger, if there are two negs. or two positives). Negative subtracted from positive = addition. Larger positive for y is 8. Only negative for x is -2. Answer is 10. Or

2. Test four cases

8 - 2 = 6
8 - (-2) = 10
3 - 2 = 1
3 - (-2) = 5

Maximum value of \(y - x\) = 10

Answer: \(1 \leq(y -x) \leq10\)

I tested all eight cases, checked my work, and time was well under a minute (so maybe what looks like the longest way isn't).

Answer E
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The largest value of y - x is 8 - (-2) = 8 + 2 = 10.

The smallest value of y - x is 3 - 2 = 1.

Thus, 1 ≤ y - x ≤ 10.

Answer: E
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