Middle School Math Quiz: Subtract Using Additive Inverse
20 questions · exam conditions
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Subtract Using Additive InverseQuestion 1 of 20

A football team gains 1212 yards, then loses 1818 yards, then gains 77 yards. Using additive inverse for each loss, what expression shows their total yardage change?

12+18+7=3712 + 18 + 7 = 37 yards gained total
12+(−18)+7=112 + (-18) + 7 = 1 yard gained total
12−18−7=−1312 - 18 - 7 = -13 yards lost total
(−12)+(−18)+(−7)=−37(-12) + (-18) + (-7) = -37 yards lost total
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Middle School Math Quiz

Middle School Math Quiz: Subtract Using Additive Inverse

Practice Subtract Using Additive Inverse in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Subtract Using Additive Inverse, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A football team gains 1212 yards, then loses 1818 yards, then gains 77 yards. Using additive inverse for each loss, what expression shows their total yardage change?

  1. 12+18+7=3712 + 18 + 7 = 37 yards gained total
  2. 12+(−18)+7=112 + (-18) + 7 = 1 yard gained total (correct answer)
  3. 12−18−7=−1312 - 18 - 7 = -13 yards lost total
  4. (−12)+(−18)+(−7)=−37(-12) + (-18) + (-7) = -37 yards lost total
Explanation: Using additive inverse, gains are positive and losses are negative. The sequence is: +12 yards (gain), -18 yards (loss), +7 yards (gain). So the expression is 12 + (-18) + 7 = 1 yard net gain. Choice B correctly applies additive inverse to the loss. Choice A treats the loss as a gain. Choice C uses subtraction notation instead of additive inverse. Choice D makes all movements negative.

Question 2

A submarine starts at a depth of −125-125 feet below sea level. It then rises 7878 feet. Using the concept that subtraction equals adding the additive inverse, which expression correctly models the submarine's final depth?

  1. −125+78=−47-125 + 78 = -47 feet, representing 4747 feet below sea level (correct answer)
  2. −125−78=−203-125 - 78 = -203 feet, representing 203203 feet below sea level
  3. −125+(−78)=−203-125 + (-78) = -203 feet, representing 203203 feet below sea level
  4. 125+(−78)=47125 + (-78) = 47 feet, representing 4747 feet above sea level
Explanation: The submarine starts at -125 feet and rises 78 feet. Rising means adding a positive value: -125 + 78 = -47 feet. The negative result indicates it's still below sea level. Choice A is correct. Choice B subtracts when it should add (rising means going up, adding positive). Choice C adds the additive inverse of 78 when it should just add 78. Choice D incorrectly starts with positive 125 instead of -125.

Question 3

A science lab starts at 15∘C15^\circ\text{C} and then the temperature drops by 20∘C20^\circ\text{C}. Which choice correctly models the situation, rewrites subtraction as addition, and gives the final temperature?

  1. 15−20=15−(−20)=−5∘C15-20=15-(-20)=-5^\circ\text{C}
  2. 15−20=15+(−20)=−5∘C15-20=15+(-20)=-5^\circ\text{C} (correct answer)
  3. 15−20=15+(−20)=5∘C15-20=15+(-20)=5^\circ\text{C}
  4. 15−20=15+20=35∘C15-20=15+20=35^\circ\text{C}
Explanation: This question tests understanding that subtraction p-q equals adding the additive inverse p+(-q), and the distance between numbers as |p-q| (absolute value of the difference). Subtraction as addition: p-q = p+(-q) by definition (subtracting q means adding the opposite -q: 5-8=5+(-8)=-3, or 7-(-3)=7+3=10 since subtracting a negative adds the positive). On the number line, p-q starts at p and moves distance |q| left if q>0 (subtracting positive), or right if q<0 (subtracting negative = adding positive); distance between p and q is |p-q|=|q-p| (absolute value of the difference, always positive: |3-10|=|-7|=7 units apart, or |10-3|=7). For example, 15-20 can be rewritten as 15+(-20)=-5 (temperature drops from 15°C to -5°C), or distance from -4 to 3: |3-(-4)|=|3+4|=7 units (or |-4-3|=|-7|=7), or 10-25 for money: 10+(-25)=-15 (debt of $15). The correct modeling for 15-20 is 15+(-20)=-5°C, as in choice B, rewriting subtraction as addition for the temperature drop. A common error is arithmetic wrong, like choice A as 15+20=35°C (adding instead of subtracting), or choice C with sum as 5°C (absolute value error), or choice D subtracting -20 which adds 20 incorrectly. Contexts: temperature 15° drops 20° (15-20=-5°, below zero), money $10 spend $25 (10-25=-15, debt), elevation 50 m descend 80 m (50-80=-30 m, below sea level 30 m); using additive inverse rewrites to addition for easier calculation.

Question 4

Calculate: -9-(-4). Use the idea that subtracting is adding the inverse.

  1. -13
  2. -5 (correct answer)
  3. 13
  4. 5
Explanation: Subtracting -4 is the same as adding its additive inverse, +4: -9 - (-4) = -9 + 4 = -5, matching choice B. Choice A (-13) comes from adding instead of subtracting the inverse, effectively computing -9-4. Choices C (13) and D (5) both ignore the negative sign on -9, treating it as if it were positive.

Question 5

You have $10 and spend $25. This can be modeled by 10−2510-25. Which statement correctly rewrites the subtraction and interprets the result?

  1. 10−25=10+25=3510-25=10+25=35, so you have $35 left.
  2. 10−25=1510-25=15, so you have $15 left.
  3. 10−25=10+(−25)=−1510-25=10+(-25)=-15, so you are $15 in debt. (correct answer)
  4. 10−25=10+(−25)=1510-25=10+(-25)=15, so you are $15 in debt.
Explanation: This question tests understanding that subtraction p - q equals adding the additive inverse p + (-q), and the distance between numbers as |p - q| (absolute value of the difference). Subtraction as addition: p - q = p + (-q) by definition (subtracting q means adding the opposite -q: 5 - 8 = 5 + (-8) = -3, or 7 - (-3) = 7 + 3 = 10 since subtracting a negative adds the positive). On the number line, p - q starts at p and moves |q| units left if q > 0 (subtracting a positive) or right if q < 0 (subtracting a negative equals adding a positive), while the distance between p and q is |p - q| = |q - p| (absolute value of the difference, always positive: |3 - 10| = |-7| = 7 units apart, or |10 - 3| = 7). For example, 15 - 20 can be rewritten as 15 + (-20) = -5 (like a temperature drop from 15°C to -5°C), or the distance from -4 to 3 is |3 - (-4)| = |3 + 4| = 7 units (or |-4 - 3| = |-7| = 7), and for money, 10 - 25 = 10 + (-25) = -15 (a debt of $15). The correct statement is 10 - 25 = 10 + (-25) = -15, so you are $15 in debt, as it rewrites properly and interprets the financial context. A common error is context misapplied, like in choice B where 10 - 25 = 10 + 25 = 35 shows positive balance instead of debt, or arithmetic wrong as in choice D where 10 + (-25) = 15 ignores the signs. Mistakes: not rewriting as addition (missing connection p - q = p + (-q)), subtracting negative as subtraction (5 - (-2) staying as subtract when should become 5 + 2 = 7), distance without absolute value (negative distance).

Question 6

Find the distance between the rational numbers 12\frac{1}{2} and −32-\frac{3}{2}. (Distance is the absolute value of their difference.)

  1. ∣12−(−32)∣=∣42∣=2\left|\frac{1}{2}-\left(-\frac{3}{2}\right)\right|=\left|\frac{4}{2}\right|=2 (correct answer)
  2. ∣12−(−32)∣=∣−42∣=−2\left|\frac{1}{2}-\left(-\frac{3}{2}\right)\right|=\left|\frac{-4}{2}\right|=-2
  3. ∣12−(−32)∣=∣22∣=1\left|\frac{1}{2}-\left(-\frac{3}{2}\right)\right|=\left|\frac{2}{2}\right|=1
  4. ∣12−(−32)∣=∣−22∣=−1\left|\frac{1}{2}-\left(-\frac{3}{2}\right)\right|=\left|\frac{-2}{2}\right|=-1
Explanation: This question tests understanding that subtraction p-q equals adding the additive inverse p+(-q), and the distance between numbers as |p-q| (absolute value of the difference). Subtraction as addition: p-q = p+(-q) by definition (subtracting q means adding the opposite -q: 5-8=5+(-8)=-3, or 7-(-3)=7+3=10 since subtracting a negative adds the positive). On the number line, p-q starts at p and moves distance |q| left if q>0 (subtracting positive), or right if q<0 (subtracting negative = adding positive); distance between p and q is |p-q|=|q-p| (absolute value of the difference, always positive: |3-10|=|-7|=7 units apart, or |10-3|=7). For example, 15-20 can be rewritten as 15+(-20)=-5 (temperature drops from 15°C to -5°C), or distance from -4 to 3: |3-(-4)|=|3+4|=7 units (or |-4-3|=|-7|=7), or 10-25 for money: 10+(-25)=-15 (debt of $15). The correct distance is |1/2 - (-3/2)| = |4/2| = 2, as in choice A, using absolute value for rational numbers. A common error is calculation mistake, like choice B with | -2/2 | = -1 (negative and wrong difference), or choice C as |2/2|=1 (incorrect sum), or choice D with | -4/2 | = -2 (wrong signs). Distance: between any two numbers p and q, calculate p-q, take absolute value |p-q| (removes sign, gives positive distance: 3-10=-7, |-7|=7 units), or reverse: |q-p| (order doesn't matter); mistakes include distance negative (without absolute value).

Question 7

A student rewrites subtraction as adding the inverse. Which expression correctly rewrites 5−85-8 as addition and gives the correct value?

  1. 5−8=5+8=135-8=5+8=13
  2. 5−8=5−(−8)=−35-8=5-(-8)=-3
  3. 5−8=5+(−8)=−35-8=5+(-8)=-3 (correct answer)
  4. 5−8=5+(−8)=35-8=5+(-8)=3
Explanation: This question tests understanding that subtraction p - q equals adding the additive inverse p + (-q), and the distance between numbers as |p - q| (absolute value of the difference). Subtraction as addition: p - q = p + (-q) by definition (subtracting q means adding the opposite -q: 5 - 8 = 5 + (-8) = -3, or 7 - (-3) = 7 + 3 = 10 since subtracting a negative adds the positive). On the number line, p - q starts at p and moves |q| units left if q > 0 (subtracting a positive) or right if q < 0 (subtracting a negative equals adding a positive), while the distance between p and q is |p - q| = |q - p| (absolute value of the difference, always positive: |3 - 10| = |-7| = 7 units apart, or |10 - 3| = 7). For example, 15 - 20 can be rewritten as 15 + (-20) = -5 (like a temperature drop from 15°C to -5°C), or the distance from -4 to 3 is |3 - (-4)| = |3 + 4| = 7 units (or |-4 - 3| = |-7| = 7), and for money, 10 - 25 = 10 + (-25) = -15 (a debt of $15). The correct rewriting is 5 - 8 = 5 + (-8) = -3, as it properly adds the additive inverse and computes the value accurately. A common error is claiming p - q = p + q instead of p + (-q), like in choice A where 5 - 8 = 5 + 8 = 13 denies the equivalence, or miscalculating the addition as in choice C where 5 + (-8) = 3, or confusing the signs as in choice D. Using the additive inverse, rewrite every subtraction as addition (p - q → p + (-q), making all operations additions), and apply addition rules (p + (-q) follows number line interpretation: start at p, move |q| left).

Question 8

The temperature difference between two cities is 23.8°F23.8°F. If City AA has a temperature of −12.3°F-12.3°F, and City BB is warmer, what is the temperature in City BB?

  1. City B is 11.5°F11.5°F, since ∣11.5−(−12.3)∣=∣23.8∣=23.8°F|11.5 - (-12.3)| = |23.8| = 23.8°F
  2. City B is 36.1°F36.1°F, since ∣−12.3+23.8∣=11.5°F|-12.3 + 23.8| = 11.5°F
  3. City B is 11.5°F11.5°F, since −12.3+23.8=11.5°F-12.3 + 23.8 = 11.5°F (correct answer)
  4. City B is −36.1°F-36.1°F, since −12.3+(−23.8)=−36.1°F-12.3 + (-23.8) = -36.1°F
Explanation: If City B is warmer and the difference is 23.8°F, then City B = City A + 23.8 = -12.3 + 23.8 = 11.5°F. We can verify: |11.5 - (-12.3)| = |11.5 + 12.3| = |23.8| = 23.8°F. Choice C correctly calculates City B's temperature. Choice A gives the right answer but had an incorrect verification calculation. Choice B has calculation errors in the verification. Choice D makes City B colder instead of warmer.

Question 9

Compute 7−(−4)7-(-4) by rewriting it as addition using the additive inverse.

  1. 7−(−4)=7+(−4)=−117-(-4)=7+(-4)=-11
  2. 7−(−4)=7+4=117-(-4)=7+4=11 (correct answer)
  3. 7−(−4)=7+(−4)=37-(-4)=7+(-4)=3
  4. 7−(−4)=7−4=37-(-4)=7-4=3
Explanation: This question tests understanding that subtraction p-q equals adding the additive inverse p+(-q), and the distance between numbers as |p-q| (absolute value of the difference). Subtraction as addition: p-q = p+(-q) by definition (subtracting q means adding the opposite -q: 5-8=5+(-8)=-3, or 7-(-3)=7+3=10 since subtracting a negative adds the positive). On the number line, p-q starts at p and moves distance |q| left if q>0 (subtracting positive), or right if q<0 (subtracting negative = adding positive); distance between p and q is |p-q|=|q-p| (absolute value of the difference, always positive: |3-10|=|-7|=7 units apart, or |10-3|=7). For example, 15-20 can be rewritten as 15+(-20)=-5 (temperature drops from 15°C to -5°C), or distance from -4 to 3: |3-(-4)|=|3+4|=7 units (or |-4-3|=|-7|=7), or 10-25 for money: 10+(-25)=-15 (debt of $15). The correct rewriting for 7-(-4) is 7+4=11, as in choice C, since subtracting negative -4 means adding positive 4. A common error is subtracting negative wrongly, like choice A as 7+(-4)=3 (keeping negative sign), or choice B as 7-4=3 (ignoring double negative), or choice D with 7+(-4)=-11 (wrong sign and sum). Using the additive inverse, rewrite every subtraction as addition (p-q→p+(-q), making all operations additions), and apply addition rules (p+(-q) follows number line: start at p, move |q| left if negative); mistakes include subtracting negative as subtraction (5-(-2) staying as subtract when should become 5+2=7). Distance concepts reinforce that order doesn't matter for |q-p|.

Question 10

Compute −6−9-6-9 by rewriting it as addition using p−q=p+(−q)p-q=p+(-q). Which is correct?

  1. −6−9=−(6−9)=3-6-9=-(6-9)=3
  2. −6−9=6+9=15-6-9=6+9=15
  3. −6−9=−6+9=3-6-9=-6+9=3
  4. −6−9=−6+(−9)=−15-6-9=-6+(-9)=-15 (correct answer)
Explanation: This question tests understanding that subtraction p - q equals adding the additive inverse p + (-q), and the distance between numbers as |p - q| (absolute value of the difference). Subtraction as addition: p - q = p + (-q) by definition (subtracting q means adding the opposite -q: 5 - 8 = 5 + (-8) = -3, or 7 - (-3) = 7 + 3 = 10 since subtracting a negative adds the positive). On the number line, p - q starts at p and moves |q| units left if q > 0 (subtracting a positive) or right if q < 0 (subtracting a negative equals adding a positive), while the distance between p and q is |p - q| = |q - p| (absolute value of the difference, always positive: |3 - 10| = |-7| = 7 units apart, or |10 - 3| = 7). For example, 15 - 20 can be rewritten as 15 + (-20) = -5 (like a temperature drop from 15°C to -5°C), or the distance from -4 to 3 is |3 - (-4)| = |3 + 4| = 7 units (or |-4 - 3| = |-7| = 7), and for money, 10 - 25 = 10 + (-25) = -15 (a debt of $15). The correct computation is -6 - 9 = -6 + (-9) = -15, as it rewrites using the additive inverse properly. A common error is arithmetic wrong, like in choice B where -6 - 9 = -6 + 9 = 3 ignores the inverse, or in choice C where it becomes 6 + 9 = 15 by changing signs incorrectly. Contexts: temperature 15° drops 20° (15 - 20 = -5°, below zero), money $10 spend $25 (10 - 25 = -15, debt), elevation 50 m descend 80 m (50 - 80 = -30 m, below sea level 30 m).

Question 11

A video game character has 1010 coins and buys an item that costs 2525 coins. This situation can be modeled by 10−2510-25. Which choice correctly rewrites the subtraction as addition and interprets the result?

  1. 10−25=10+(−25)=−1510-25=10+(-25)=-15, so the character is short 1515 coins. (correct answer)
  2. 10−25=10+(−25)=1510-25=10+(-25)=15, so the character has 1515 coins left.
  3. 10−25=10+25=3510-25=10+25=35, so the character has 3535 coins left.
  4. 10−25=10+(−25)=−3510-25=10+(-25)=-35, so the character is short 3535 coins.
Explanation: Rewriting subtraction as addition: 10-25=10+(-25). Adding 10 and -25 (different signs, so subtract magnitudes and keep the sign of the larger magnitude): 25-10=15, and since -25 has the larger magnitude, the result is -15, so the character is short 15 coins, matching choice A. Choice B rewrites correctly as 10+(-25) but makes an arithmetic sign error in the final sum, giving +15 instead of -15. Choice C forgets to make 25 negative when rewriting, effectively computing 10+25 instead of 10+(-25). Choice D rewrites correctly as 10+(-25) but makes an arithmetic slip, giving -35 instead of -15.

Question 12

An elevator is at -3 (3 floors below ground level) and goes up to floor 5. What is the distance traveled? (Use |p-q|.)

  1. |5-(-3)| = |8| = 8 (correct answer)
  2. |-3-5| = -8
  3. |-3+5| = |2| = 2
  4. 5-(-3) = -8
Explanation: The distance between two points is the absolute value of their difference, |p-q|. From -3 to 5, the distance is |5-(-3)| = |8| = 8 floors, matching Choice A. Choice B computes the value inside the bars correctly as -8, but then leaves off the absolute value, giving a negative number, and distance can never be negative. Choice C mistakenly adds the two positions instead of subtracting them, giving the wrong value entirely. Choice D makes a sign error, since 5-(-3) actually equals 5+3=8, not -8.

Question 13

Which expression gives the distance between 10 and 3 on a number line, and what is that distance?

  1. |10-3| = -7, so the distance is -7
  2. 3-10 = 7, so the distance is 7
  3. |10-3| = |13| = 13, so the distance is 13
  4. |3-10| = |-7| = 7, so the distance is 7 (correct answer)
Explanation: The distance between two points on a number line is the absolute value of their difference, |p-q|. Using p=3 and q=10, the distance is |3-10| = |-7| = 7, matching Choice D. Choice A finds the correct value inside the bars, 10-3=7, but drops the absolute value, incorrectly stating the answer as -7, which cannot be a distance. Choice B makes an arithmetic mistake: 3-10 actually equals -7, not 7, so this statement's own arithmetic is wrong. Choice C makes an error inside the absolute value bars, computing 10-3 as if it were 10+3, giving 13 instead of 7.

Question 14

Compare distances on the number line: Which is greater, the distance between −2-2 and 77 or the distance between −5-5 and 11?

  1. They are equal because both differences are 99.
  2. Distance between −5-5 and 11 is greater because ∣−5−1∣=9|-5-1|=9.
  3. Distance between −5-5 and 11 is greater because ∣1−(−5)∣=−6|1-(-5)|=-6.
  4. Distance between −2-2 and 77 is greater because ∣7−(−2)∣=9|7-(-2)|=9 and ∣1−(−5)∣=6|1-(-5)|=6. (correct answer)
Explanation: This question tests understanding that subtraction p - q equals adding the additive inverse p + (-q), and the distance between numbers as |p - q| (absolute value of the difference). Subtraction as addition: p - q = p + (-q) by definition (subtracting q means adding the opposite -q: 5 - 8 = 5 + (-8) = -3, or 7 - (-3) = 7 + 3 = 10 since subtracting a negative adds the positive). On the number line, p - q starts at p and moves |q| units left if q > 0 (subtracting a positive) or right if q < 0 (subtracting a negative equals adding a positive), while the distance between p and q is |p - q| = |q - p| (absolute value of the difference, always positive: |3 - 10| = |-7| = 7 units apart, or |10 - 3| = 7). For example, 15 - 20 can be rewritten as 15 + (-20) = -5 (like a temperature drop from 15°C to -5°C), or the distance from -4 to 3 is |3 - (-4)| = |3 + 4| = 7 units (or |-4 - 3| = |-7| = 7), and for money, 10 - 25 = 10 + (-25) = -15 (a debt of $15). The distance between -2 and 7 is greater because |7 - (-2)| = 9 and |1 - (-5)| = 6, correctly comparing the absolute differences. A common error is miscalculating the distance, like in choice C where |1 - (-5)| = -6 claims a negative, or in choice D where |-5 - 1| = 9 is arithmetically wrong since it's 6. Distance: between any two numbers p and q, calculate p - q and take the absolute value |p - q| (removes the sign, gives positive distance: 3 - 10 = -7, |-7| = 7 units), or reverse |q - p| (order doesn't matter for distance, both give the same).

Question 15

Which statement correctly verifies the equivalence p-q=p+(-q) for p=-2 and q=7?

  1. -2-7=-2+(-7) (correct answer)
  2. -2-7=-2+7
  3. -2-7=-(2-7)
  4. -2-7=2+7
Explanation: To verify p-q=p+(-q), substitute p=-2 and q=7 into both sides: p-q = -2-7 = -9, and p+(-q) = -2+(-7) = -9. Since both sides equal -9, this is a true statement showing the equivalence, matching choice A. Choice B (-2-7=-2+7) is false, since -2+7=5, not -9 - it mistakenly adds +7 instead of the additive inverse -7. Choices C and D both use incorrect sign manipulations that don't represent the additive-inverse rule being tested.

Question 16

A hiker is at an elevation of −12-12 meters (below sea level) and then goes down 77 more meters. This can be modeled by −12−7-12-7. What is the new elevation? (Rewrite subtraction as addition.)

  1. −12−7=−12+(−7)=−19-12-7=-12+(-7)=-19 meters (correct answer)
  2. −12−7=−12−(−7)=−19-12-7=-12-(-7)=-19 meters
  3. −12−7=−12+7=−5-12-7=-12+7=-5 meters
  4. −12−7=−12+(−7)=19-12-7=-12+(-7)=19 meters
Explanation: This question tests understanding that subtraction p-q equals adding the additive inverse p+(-q), and the distance between numbers as |p-q| (absolute value of the difference). Subtraction as addition: p-q = p+(-q) by definition (subtracting q means adding the opposite -q: 5-8=5+(-8)=-3, or 7-(-3)=7+3=10 since subtracting a negative adds the positive). On the number line, p-q starts at p and moves distance |q| left if q>0 (subtracting positive), or right if q<0 (subtracting negative = adding positive); distance between p and q is |p-q|=|q-p| (absolute value of the difference, always positive: |3-10|=|-7|=7 units apart, or |10-3|=7). For example, 15-20 can be rewritten as 15+(-20)=-5 (temperature drops from 15°C to -5°C), or distance from -4 to 3: |3-(-4)|=|3+4|=7 units (or |-4-3|=|-7|=7), or 10-25 for money: 10+(-25)=-15 (debt of $15). The correct rewriting for -12-7 is -12+(-7)=-19 meters, as in choice B, modeling descent below sea level. A common error is sign mishandling, like choice A as -12+7=-5 (adding positive), or choice C subtracting -7 which adds 7, or choice D with positive 19 (ignoring signs). Contexts: elevation 50 m descend 80 m (50-80=-30 m, below sea level 30 m); using additive inverse rewrites to addition (p-q→p+(-q)), apply rules (start at p, move left). Mistakes include arithmetic wrong (like -12-7=19).

Question 17

Compare distances on the number line: Which distance is greater? Distance 1: between −2-2 and 66. Distance 2: between 33 and −5-5.

  1. Distance 1 is greater
  2. They are equal (correct answer)
  3. Distance 2 is greater
  4. Not enough information
Explanation: This question tests understanding that subtraction p-q equals adding the additive inverse p+(-q), and the distance between numbers as |p-q| (absolute value of the difference). Subtraction as addition: p-q = p+(-q) by definition (subtracting q means adding the opposite -q: 5-8=5+(-8)=-3, or 7-(-3)=7+3=10 subtracting negative adds positive). Number line: p-q starts at p, moves distance |q| left if q>0 (subtracting positive), or right if q<0 (subtracting negative = adding positive). Distance between p and q: |p-q|=|q-p| (absolute value of difference, always positive: |3-10|=|-7|=7 units apart, or |10-3|=7). Distance 1: |-2-6|=|-8|=8; Distance 2: |3-(-5)|=|8|=8, so they are equal, matching choice C. A common error is thinking order matters without absolute value, leading to A or B. Distance: between any two numbers p and q, calculate p-q, take absolute value |p-q| (removes sign, gives positive distance: 3-10=-7, |-7|=7 units), or reverse: |q-p| (order doesn't matter for distance, both give same). Mistakes: not rewriting as addition (missing connection p-q=p+(-q)), subtracting negative as subtraction (5-(-2) staying as subtract when should become 5+2=7), distance without absolute value (negative distance).

Question 18

Which equation correctly shows that subtracting a negative is the same as adding a positive?

  1. 5−(−2)=(−5)+2=−35-(-2)=(-5)+2=-3
  2. 5−(−2)=5−2=35-(-2)=5-2=3
  3. 5−(−2)=5+2=75-(-2)=5+2=7 (correct answer)
  4. 5−(−2)=5+(−2)=35-(-2)=5+(-2)=3
Explanation: This question tests understanding that subtraction p - q equals adding the additive inverse p + (-q), and the distance between numbers as |p - q| (absolute value of the difference). Subtraction as addition: p - q = p + (-q) by definition (subtracting q means adding the opposite -q: 5 - 8 = 5 + (-8) = -3, or 7 - (-3) = 7 + 3 = 10 since subtracting a negative adds the positive). On the number line, p - q starts at p and moves |q| units left if q > 0 (subtracting a positive) or right if q < 0 (subtracting a negative equals adding a positive), while the distance between p and q is |p - q| = |q - p| (absolute value of the difference, always positive: |3 - 10| = |-7| = 7 units apart, or |10 - 3| = 7). For example, 15 - 20 can be rewritten as 15 + (-20) = -5 (like a temperature drop from 15°C to -5°C), or the distance from -4 to 3 is |3 - (-4)| = |3 + 4| = 7 units (or |-4 - 3| = |-7| = 7), and for money, 10 - 25 = 10 + (-25) = -15 (a debt of $15). The correct equation is 5 - (-2) = 5 + 2 = 7, as it shows subtracting a negative becomes adding a positive. A common error is subtracting negative wrong, like in choice A where 5 - (-2) = 5 + (-2) = 3 treats it as adding negative, or in choice B where it stays as 5 - 2 = 3 without converting. Using the additive inverse, rewrite every subtraction as addition (p - q → p + (-q), making all operations additions), and apply addition rules (p + (-q) follows number line interpretation: start at p, move |q| left).

Question 19

Two students argue about distance: Student 1 says the distance from 33 to 1010 is ∣10−3∣=7|10-3|=7. Student 2 says the distance from 1010 to 33 is ∣3−10∣=−7|3-10|=-7. Which statement is correct?​

  1. Both are incorrect because distance depends on the order of subtraction.
  2. Student 2 is correct because 3−10=−73-10=-7.
  3. Student 1 is correct, and Student 2 should have ∣3−10∣=∣−7∣=7|3-10|=|-7|=7. (correct answer)
  4. Both are correct because distance can be negative.
Explanation: This question tests understanding that subtraction p - q equals adding the additive inverse p + (-q), and the distance between numbers as |p - q| (absolute value of the difference). Subtraction as addition: p - q = p + (-q) by definition (subtracting q means adding the opposite -q: 5 - 8 = 5 + (-8) = -3, or 7 - (-3) = 7 + 3 = 10 since subtracting a negative adds the positive). On the number line, p - q starts at p and moves |q| units left if q > 0 (subtracting a positive) or right if q < 0 (subtracting a negative equals adding a positive), while the distance between p and q is |p - q| = |q - p| (absolute value of the difference, always positive: |3 - 10| = |-7| = 7 units apart, or |10 - 3| = 7). For example, 15 - 20 can be rewritten as 15 + (-20) = -5 (like a temperature drop from 15°C to -5°C), or the distance from -4 to 3 is |3 - (-4)| = |3 + 4| = 7 units (or |-4 - 3| = |-7| = 7), and for money, 10 - 25 = 10 + (-25) = -15 (a debt of $15). The correct statement is that Student 1 is right, and Student 2 should have |3 - 10| = |-7| = 7, as distance uses absolute value and is always positive regardless of order. A common error is claiming distance can be negative, like in choice A where Student 2 is correct with -7, or in choice D where both are correct with negative distance. Distance: between any two numbers p and q, calculate p - q and take the absolute value |p - q| (removes the sign, gives positive distance: 3 - 10 = -7, |-7| = 7 units), or reverse |q - p| (order doesn't matter for distance, both give the same).

Question 20

The distance between two numbers on a number line is the absolute value of their difference. What is the distance between −4-4 and 33?​

  1. ∣−4−3∣=−7|-4-3|=-7, so the distance is −7-7
  2. ∣3−(−4)∣=∣7∣=7|3-(-4)|=|7|=7 (correct answer)
  3. 3−(−4)=−13-(-4)= -1, so the distance is −1-1
  4. ∣−4−3∣=∣−7∣=−7|-4-3|=|-7|=-7
Explanation: This question tests understanding that subtraction p - q equals adding the additive inverse p + (-q), and the distance between numbers as |p - q| (absolute value of the difference). Subtraction as addition: p - q = p + (-q) by definition (subtracting q means adding the opposite -q: 5 - 8 = 5 + (-8) = -3, or 7 - (-3) = 7 + 3 = 10 since subtracting a negative adds the positive). On the number line, p - q starts at p and moves |q| units left if q > 0 (subtracting a positive) or right if q < 0 (subtracting a negative equals adding a positive), while the distance between p and q is |p - q| = |q - p| (absolute value of the difference, always positive: |3 - 10| = |-7| = 7 units apart, or |10 - 3| = 7). For example, 15 - 20 can be rewritten as 15 + (-20) = -5 (like a temperature drop from 15°C to -5°C), or the distance from -4 to 3 is |3 - (-4)| = |3 + 4| = 7 units (or |-4 - 3| = |-7| = 7), and for money, 10 - 25 = 10 + (-25) = -15 (a debt of $15). The correct distance is |3 - (-4)| = |7| = 7, as it applies the absolute value to the difference properly. A common error is treating distance as negative without absolute value, like in choice A where |-4 - 3| = |-7| = -7 claims a negative distance, or in choice C where 3 - (-4) = -1 miscalculates the subtraction. Distance: between any two numbers p and q, calculate p - q and take the absolute value |p - q| (removes the sign, gives positive distance: 3 - 10 = -7, |-7| = 7 units), or reverse |q - p| (order doesn't matter for distance, both give the same).