Question

which of the following is another way to express the difference between -5 and 8?
-5 + 8
-5 - (-8)
-5 + (-8)
8 - 5

Explanation:

Step1: Recall the definition of difference

The difference between two numbers \(a\) and \(b\) is \(a - b\). So the difference between \(-5\) and \(8\) is \(-5-8\)? Wait, no, wait. Wait, the difference between \(a\) and \(b\) can be \(a - b\) or \(b - a\), but let's check the options. Wait, first, let's recall the rule of subtraction: \(a - b=a+(-b)\). So the difference between \(-5\) and \(8\) is \(-5 - 8\)? No, wait, no. Wait, the problem says "the difference between -5 and 8". The difference between two numbers \(x\) and \(y\) is \(x - y\) or \(y - x\). But let's check the options. Let's analyze each option:

Option 1: \(-5 + 8\). Let's recall that \(a - b=a+(-b)\), but here, if we consider the difference between \(-5\) and \(8\) as \(-5-8\)? No, wait, no. Wait, maybe I made a mistake. Wait, the difference between \(-5\) and \(8\) is \(-5 - 8\) or \(8-(-5)\)? Wait, no. Wait, the difference between two numbers \(x\) and \(y\) is \(x - y\). So if \(x=-5\) and \(y = 8\), then the difference is \(-5-8\). But let's check the options. Wait, maybe the problem is about the difference as in \(x - y\) where \(x=-5\) and \(y = 8\), or maybe \(y - x\). Wait, let's check the options:

Option 1: \(-5 + 8\). Let's simplify each option:

Option 1: \(-5 + 8=3\)

Option 2: \(-5-(-8)=-5 + 8 = 3\)

Option 3: \(-5+(-8)=-13\)

Option 4: \(8 - 5 = 3\)

Wait, that can't be. Wait, maybe the problem is about the difference as in \(-5 - 8\) or \(8-(-5)\)? No, wait, maybe I misread the problem. Wait, the problem says "the difference between -5 and 8". So the difference can be \(-5 - 8\) or \(8 - (-5)\)? Wait, no. Wait, the difference between two numbers \(a\) and \(b\) is \(a - b\). So if \(a=-5\) and \(b = 8\), then the difference is \(-5 - 8=-13\). But none of the options give \(-13\) except option 3, but option 3 is \(-5+(-8)=-13\), which is \(-5 - 8\). But wait, the options have \(-5 + 8\), \(-5-(-8)\), \(-5+(-8)\), \(8 - 5\). Wait, maybe the problem is about the difference as in \(8-(-5)\)? No, that would be \(8 + 5 = 13\). Wait, maybe I made a mistake. Wait, let's re-express each option:

Option 1: \(-5 + 8\). This is equivalent to \(-5 - (-8)\) (because subtracting a negative is adding the positive). Wait, no: \(-5 - (-8)=-5 + 8\), which is the same as option 1. Wait, option 2 is \(-5-(-8)\), which is equal to \(-5 + 8\), same as option 1. Wait, that can't be. Wait, maybe the problem is about the difference between -5 and 8 as in \(-5 - 8\) or \(8 - (-5)\)? No, that doesn't match. Wait, maybe the problem is using "difference" as in the result of subtracting, but maybe the question is phrased as "the difference between -5 and 8" meaning \(-5 - 8\) or \(8 - (-5)\), but the options don't have \(8 - (-5)\). Wait, let's check the options again:

Option 1: \(-5 + 8\)

Option 2: \(-5-(-8)\)

Option 3: \(-5+(-8)\)

Option 4: \(8 - 5\)

Wait, let's compute each:

Option 1: \(-5 + 8 = 3\)

Option 2: \(-5 - (-8)=-5 + 8 = 3\)

Option 3: \(-5 + (-8)=-13\)

Option 4: \(8 - 5 = 3\)

Wait, so options 1,2,4 give 3, option 3 gives -13. But the difference between -5 and 8 is \(-5 - 8=-13\) or \(8 - (-5)=13\). Wait, this is confusing. Maybe the problem is about the difference as in \(8 - (-5)\)? No, that's 13. But the options have 3. Wait, maybe the problem is a typo, or maybe I misinterpret "difference between -5 and 8". Maybe it's the difference between 8 and -5? Then the difference is \(8 - (-5)=13\), but that's not in the options. Wait, no. Wait, maybe the problem is asking for an equivalent expression for \(-5 - 8\)? No, option 3 is \(-5+(-8)=-13\), which is \(-5 - 8\). But the options 1,2,4 give 3. Wait, maybe the problem is about the difference as in \(-5 - (-8)\)? No, that's \(-5 + 8 = 3\). Wait, maybe the question is "the difference between -5 and 8" as in \(-5 - 8\) or \(8 - (-5)\), but the options are different. Wait, maybe the original problem is "the difference between -5 and 8" meaning \(-5 - 8\), but option 3 is \(-5+(-8)=-13\), which is correct. But the other options give 3. This is confusing. Wait, maybe I made a mistake in the sign. Let's re-express each option:

Option 1: \(-5 + 8\) is the same as \(8 - 5\) (commutative property of addition). Option 2: \(-5 - (-8)=-5 + 8\) (because subtracting a negative is adding the positive). Option 3: \(-5 + (-8)=-13\) (adding a negative is subtracting the positive). Option 4: \(8 - 5 = 3\). So options 1,2,4 are equivalent to \(3\), option 3 is \(-13\). Now, the difference between -5 and 8: if we take "difference" as in \(a - b\) where \(a=-5\) and \(b = 8\), then it's \(-5 - 8=-13\), which is option 3. But if we take "difference" as in \(b - a\) where \(a=-5\) and \(b = 8\), then it's \(8 - (-5)=13\), which is not in the options. Wait, this is a problem. Wait, maybe the question is "the difference between -5 and 8" as in \(-5 - 8\), but option 3 is \(-5+(-8)=-13\), which is correct. But the other options give 3. Wait, maybe the question is a mistake, or maybe I misread. Wait, let's check the options again. The options are:

- \(-5 + 8\)

- \(-5 - (-8)\)

- \(-5 + (-8)\)

- \(8 - 5\)

Wait, maybe the problem is about the difference between -5 and 8 as in \(-5 - 8\), which is \(-5+(-8)\), so option 3. But then why do options 1,2,4 give 3? Wait, maybe the problem is "the difference between -5 and 8" as in \(8 - (-5)\)? No, that's 13. Wait, I'm confused. Wait, let's check the arithmetic again:

- \(-5 + 8 = 3\)

- \(-5 - (-8) = -5 + 8 = 3\)

- \(-5 + (-8) = -13\)

- \(8 - 5 = 3\)

So three options give 3, one gives -13. But the difference between -5 and 8 is either \(-5 - 8 = -13\) or \(8 - (-5) = 13\). So none of the options give 13, but option 3 gives -13, which is \(-5 - 8\). So maybe the question is asking for \(-5 - 8\), which is equivalent to \(-5 + (-8)\), so option 3. But that seems odd. Wait, maybe the problem is phrased as "the difference between -5 and 8" meaning \(-5 - 8\), so the answer is option 3: \(-5 + (-8)\). But let's confirm.

Wait, maybe the original problem is "the difference between -5 and 8" as in subtracting 8 from -5, which is \(-5 - 8 = -13\), and \(-5 + (-8) = -13\), so that's the equivalent expression. So the correct option is \(-5 + (-8)\), which is option 3. But wait, the other options give 3. This is conflicting. Wait, maybe I made a mistake in the sign of the difference. Let's check with a number line. The difference between -5 and 8: the distance between -5 and 8 is \(|8 - (-5)| = 13\), but the difference (not absolute) is \(8 - (-5) = 13\) or \(-5 - 8 = -13\). So the non-absolute difference is either 13 or -13. The options have -13 (option 3) and 3 (options 1,2,4). So if the problem is asking for the difference (not absolute), then option 3 is correct. So the answer is \(-5 + (-8)\), which is option 3.

Wait, but let's check the options again. The options are:

A. \(-5 + 8\)

B. \(-5 - (-8)\)

C. \(-5 + (-8)\)

D. \(8 - 5\)

So the correct answer is C. \(-5 + (-8)\)

Answer:

C. \(-5 + (-8)\)