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To determine which description fits the sum [tex]\( p + (-q) \)[/tex] where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are positive numbers, let's go through the detailed reasoning:
1. Identify the basics of the operation:
- When we have [tex]\( p \)[/tex] and [tex]\( q \)[/tex] as positive numbers, [tex]\( p \)[/tex] is a positive number and [tex]\( q \)[/tex] is also a positive number.
- The expression [tex]\( (-q) \)[/tex] means we are taking the negative of [tex]\( q \)[/tex].
2. Understanding the sum [tex]\( p + (-q) \)[/tex]:
- Summing [tex]\( p \)[/tex] and [tex]\( -q \)[/tex] mathematically translates to [tex]\( p - q \)[/tex]. This is subtracting [tex]\( q \)[/tex] from [tex]\( p \)[/tex].
3. Interpreting the result:
- The operation [tex]\( p - q \)[/tex] results in a number which is exactly [tex]\( q \)[/tex] units to the left (in the negative direction) on the number line from [tex]\( p \)[/tex].
- This happens because subtracting [tex]\( q \)[/tex] is like moving [tex]\( q \)[/tex] spaces to the left on the number line starting from [tex]\( p \)[/tex].
4. Verifying with values:
- As an example, let [tex]\( p = 5 \)[/tex] and [tex]\( q = 3 \)[/tex]:
- [tex]\( p - q = 5 - 3 = 2 \)[/tex].
- This confirms that moving [tex]\( q \)[/tex] units (which is 3 units) to the left of [tex]\( p \)[/tex] (which is 5) lands us at 2 on the number line.
5. Conclusion from provided descriptions:
- First option: "The sum [tex]\( p + (-q) \)[/tex] is the number located a distance [tex]\( |p| \)[/tex] from [tex]\( q \)[/tex] in the negative direction."
- This is incorrect. It should involve [tex]\( p \)[/tex], not [tex]\( q \)[/tex], as the starting point.
- Second option: "The sum [tex]\( p + (-q) \)[/tex] is the number located a distance [tex]\( |q| \)[/tex] from [tex]\( p \)[/tex] in the negative direction."
- This is correct because it accurately describes the result of [tex]\( p - q \)[/tex] as the number [tex]\( q \)[/tex] units to the left of [tex]\( p \)[/tex].
- Third option: "The sum [tex]\( p + (-q) \)[/tex] is the number located a distance [tex]\( |q| \)[/tex] from [tex]\( p \)[/tex] in the positive direction."
- This is incorrect because it mentions moving in the positive direction rather than the negative direction.
- Fourth option: "The sum [tex]\( p + (-q) \)[/tex] is the number located a distance [tex]\( |p| \)[/tex] from [tex]\( d \)[/tex] in the positive direction."
- This option contains an apparent error in referencing [tex]\( d \)[/tex] and is incorrect overall.
Hence, the correct description for the sum [tex]\( p + (-q) \)[/tex] where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are positive numbers is:
The sum [tex]\( p + (-q) \)[/tex] is the number located a distance [tex]\( |q| \)[/tex] from [tex]\( p \)[/tex] in the negative direction.
1. Identify the basics of the operation:
- When we have [tex]\( p \)[/tex] and [tex]\( q \)[/tex] as positive numbers, [tex]\( p \)[/tex] is a positive number and [tex]\( q \)[/tex] is also a positive number.
- The expression [tex]\( (-q) \)[/tex] means we are taking the negative of [tex]\( q \)[/tex].
2. Understanding the sum [tex]\( p + (-q) \)[/tex]:
- Summing [tex]\( p \)[/tex] and [tex]\( -q \)[/tex] mathematically translates to [tex]\( p - q \)[/tex]. This is subtracting [tex]\( q \)[/tex] from [tex]\( p \)[/tex].
3. Interpreting the result:
- The operation [tex]\( p - q \)[/tex] results in a number which is exactly [tex]\( q \)[/tex] units to the left (in the negative direction) on the number line from [tex]\( p \)[/tex].
- This happens because subtracting [tex]\( q \)[/tex] is like moving [tex]\( q \)[/tex] spaces to the left on the number line starting from [tex]\( p \)[/tex].
4. Verifying with values:
- As an example, let [tex]\( p = 5 \)[/tex] and [tex]\( q = 3 \)[/tex]:
- [tex]\( p - q = 5 - 3 = 2 \)[/tex].
- This confirms that moving [tex]\( q \)[/tex] units (which is 3 units) to the left of [tex]\( p \)[/tex] (which is 5) lands us at 2 on the number line.
5. Conclusion from provided descriptions:
- First option: "The sum [tex]\( p + (-q) \)[/tex] is the number located a distance [tex]\( |p| \)[/tex] from [tex]\( q \)[/tex] in the negative direction."
- This is incorrect. It should involve [tex]\( p \)[/tex], not [tex]\( q \)[/tex], as the starting point.
- Second option: "The sum [tex]\( p + (-q) \)[/tex] is the number located a distance [tex]\( |q| \)[/tex] from [tex]\( p \)[/tex] in the negative direction."
- This is correct because it accurately describes the result of [tex]\( p - q \)[/tex] as the number [tex]\( q \)[/tex] units to the left of [tex]\( p \)[/tex].
- Third option: "The sum [tex]\( p + (-q) \)[/tex] is the number located a distance [tex]\( |q| \)[/tex] from [tex]\( p \)[/tex] in the positive direction."
- This is incorrect because it mentions moving in the positive direction rather than the negative direction.
- Fourth option: "The sum [tex]\( p + (-q) \)[/tex] is the number located a distance [tex]\( |p| \)[/tex] from [tex]\( d \)[/tex] in the positive direction."
- This option contains an apparent error in referencing [tex]\( d \)[/tex] and is incorrect overall.
Hence, the correct description for the sum [tex]\( p + (-q) \)[/tex] where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are positive numbers is:
The sum [tex]\( p + (-q) \)[/tex] is the number located a distance [tex]\( |q| \)[/tex] from [tex]\( p \)[/tex] in the negative direction.
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