Subtract Using Additive Inverse - 7th Grade Math

1. Find the distance between the numbers -4 and 3 on a number line. (Use absolute value of the difference.)
   Answer: |-4-3|=|-7|=7 The distance between two points on a number line is the absolute value of their difference, |p - q|. Using p = -4 and q = 3, the distance is |-4 - 3| = |-7| = 7, matching Choice C. Choice A leaves off the absolute value bars entirely, giving a negative result of -7, but distance can never be negative. Choice B computes the difference in the other order but also drops the absolute value, again giving -7 instead of a positive distance. Choice D makes a sign error inside the absolute value bars, turning 3 - (-4) into 3 - 4, which gives an incorrect result of 1.
2. Compute $7-(-4)$ by rewriting it as addition using the additive inverse.
   Answer: $7-(-4)=7+4=11$ 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|.
3. Rewrite the subtraction expression as an addition expression using the additive inverse: $5-8$.
   Answer: $5+(-8)$ Rewriting a subtraction expression as addition means adding the additive inverse of the number being subtracted, so 5 minus 8 becomes 5 plus negative 8, matching choice D. Choice A, negative 5 plus 8, incorrectly flips the sign of the first number instead of the second. Choice B, negative 5 minus 8, is still written as a subtraction and does not represent the additive inverse rule being tested. Choice C, 5 plus 8, adds positive 8 instead of its opposite, negative 8, which changes the value of the expression entirely. The additive inverse rule always keeps the first number the same and changes only the sign of the number being subtracted.
4. Two points on a number line are at -4 and 3. What is the distance between them? (Use absolute value of a difference.)
   Answer: |3-(-4)|=|7|=7 The distance between two points on a number line is the absolute value of their difference, |p - q|. Using p = 3 and q = -4, the distance is |3 - (-4)| = |7| = 7, matching Choice D. Choice A makes a sign error when removing the parentheses, turning 3-(-4) into 3-4, which gives an incorrect result of 1. Choice B correctly finds 7 inside the absolute value bars but then reports the answer as -7, forgetting that absolute value results are always positive. Choice C makes an arithmetic mistake, since -4-3 actually equals -7, not -1, leading to the wrong final answer.
5. Calculate: -9-(-4). Use the idea that subtracting is adding the inverse.
   Answer: -5 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.
6. You have $10 and you spend $25. Your new balance can be found by $10 - 25$. What is the result?
   Answer: $-15$ 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 $|10-3|=7$). For $10-25$, rewrite as $10 + (-25)=-15$, matching choice C. A common error is choosing A ($15$), which might ignore the signs, or D ($35$), which adds instead. Using the additive inverse: rewrite every subtraction as addition ($p-q \to p + (-q)$, makes all operations additions), apply addition rules ($p + (-q)$ follows number line interpretation: start $p$, move $|q|$ left). Contexts: temperature $15^\circ$ drops $20^\circ$ ($15-20=-5^\circ$, 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).
7. Find the distance between the rational numbers $\frac{1}{2}$ and $-\frac{3}{2}$. (Distance is the absolute value of their difference.)
   Answer: $\left|\frac{1}{2}-\left(-\frac{3}{2}\right)\right|=\left|\frac{4}{2}\right|=2$ 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).
8. A science lab records a change in pH as $\frac{3}{4}-\left(\frac{5}{4}\right)$. Which option correctly rewrites and evaluates it using $p-q=p+(-q)$?
   Answer: $\frac{3}{4}+\left(-\frac{5}{4}\right)=\frac{-2}{4}=-\frac{1}{2}$ 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). For (3/4) - (5/4), rewrite as (3/4) + (-5/4) = -2/4 = -1/2, matching choice B. A common error is choosing A, which adds positive 5/4 instead, or C, which doesn't use the inverse. Using the additive inverse: rewrite every subtraction as addition (p-q→p+(-q), makes all operations additions), apply addition rules (p+(-q) follows number line interpretation: start p, move |q| left). 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).
9. On a number line, Maria moves from position $\frac{3}{4}$ to position $-\frac{5}{8}$. Using subtraction as adding the additive inverse, what is her displacement?
   Answer: $-\frac{5}{8} - \frac{3}{4} = -\frac{5}{8} + (-\frac{6}{8}) = -\frac{11}{8}$ units When you encounter displacement problems on a number line, you need to calculate the change in position using the formula: final position minus initial position. This tells you both the distance and direction of movement. Maria starts at $\frac{3}{4}$ and ends at $-\frac{5}{8}$. Her displacement is: $-\frac{5}{8} - \frac{3}{4}$. To subtract fractions, you need a common denominator. Converting $\frac{3}{4}$ to eighths: $\frac{3}{4} = \frac{6}{8}$. Now calculate: $-\frac{5}{8} - \frac{6}{8} = -\frac{5}{8} + (-\frac{6}{8}) = -\frac{11}{8}$. The negative sign indicates Maria moved left on the number line, which makes sense since she went from a positive position to a negative one. Choice A incorrectly adds the absolute values of both positions, ignoring direction entirely. Choice B calculates $-\frac{5}{8} + \frac{3}{4}$, which reverses the displacement formula—this would be the calculation if Maria moved from $-\frac{5}{8}$ to $\frac{3}{4}$ instead. Choice C uses $\frac{3}{4} - (-\frac{5}{8})$, which calculates the distance between the points but gets the direction wrong by subtracting in the wrong order. Remember: displacement equals final position minus initial position. The order matters because it determines whether your answer is positive (rightward movement) or negative (leftward movement). Always double-check that your answer's sign matches the actual direction of movement on the number line.
10. On a coordinate plane, point $A$ is at $(-3.2, 4.7)$ and point $B$ is at $(1.8, 4.7)$. Using the absolute value of the difference, what is the distance between points $A$ and $B$?
    Answer: $|(-3.2) - 1.8| = |-5.0| = 5.0$ units horizontally Since points A and B share the same y-coordinate (4.7), they lie on the same horizontal line, so the distance is the absolute value of the difference of their x-coordinates: |(-3.2)-1.8|=|-5.0|=5.0 units, matching choice D. Choice A adds the x-coordinates instead of subtracting them. Choice B mistakenly uses the y-coordinates, which are equal, and wrongly concludes the points are identical even though their x-coordinates differ. Choice C subtracts the absolute values of the coordinates instead of taking the absolute value of their difference, which is not the same operation and gives a negative result that can't represent a distance.
11. A hiker is at an elevation of $-12$ meters (below sea level) and then goes down $7$ more meters. This can be modeled by $-12-7$. What is the new elevation? (Rewrite subtraction as addition.)
    Answer: $-12-7=-12+(-7)=-19$ meters 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).
12. Two students argue about distance: Student 1 says the distance from $3$ to $10$ is $|10-3|=7$. Student 2 says the distance from $10$ to $3$ is $|3-10|=-7$. Which statement is correct?​
    Answer: Student 1 is correct, and Student 2 should have $|3-10|=|-7|=7$. 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).
13. The distance between two numbers on a number line is the absolute value of their difference. What is the distance between $-4$ and $3$?
    Answer: $|3-(-4)|=|7|=7$ 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).
14. A student rewrites subtraction as adding the inverse. Which expression correctly rewrites $5-8$ as addition and gives the correct value?
    Answer: $5-8=5+(-8)=-3$ 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).
15. Rewrite the subtraction as addition using the additive inverse, and then find the value: $5-8$.
    Answer: $5-8=5+(-8)=-3$ 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 5-8 is 5+(-8)=-3, as in choice B, which properly applies the additive inverse and calculates the sum. A common error is confusing the signs, like in choice A where 5-8=5+8=13 incorrectly adds positive 8 instead of -8, or choice C with the wrong sum of -3 as 3, or choice D subtracting -8 which would be adding 8. 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). Mistakes include not rewriting as addition (missing the connection p-q=p+(-q)), or handling subtracting negative incorrectly (like 5-(-2) should become 5+2=7, not staying as subtraction).
16. Two students argue about distance: Student 1 says the distance from $3$ to $10$ is $|10-3|=7$. Student 2 says the distance from $10$ to $3$ is $|3-10|=-7$. Which statement is correct?
    Answer: Student 1 is correct, and Student 2 should have $|3-10|=|-7|=7$. 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).
17. Compute $-6-9$ by rewriting it as addition using $p-q=p+(-q)$. Which is correct?
    Answer: $-6-9=-6+(-9)=-15$ 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).
18. An underwater robot is at an elevation of $-\tfrac{3}{2}$ meters (below sea level). It then moves down another $\tfrac{5}{2}$ meters. Which expression gives its new elevation?
    Answer: $-\tfrac{3}{2}-\tfrac{5}{2}=-\tfrac{8}{2}=-4$ 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 expression is -3/2 - 5/2 = -8/2 = -4, as it computes the new elevation after descending. A common error is rewriting incorrectly, like in choice A where -3/2 - 5/2 = -3/2 + 5/2 = 1 adds instead of subtracting, or in choice C where it becomes positive 4 without proper 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).
19. In science class, the temperature starts at $15^\circ\text{C}$ and then drops $20^\circ\text{C}$. Which shows the subtraction as addition and the correct final temperature?
    Answer: $15-20=15+(-20)=-5^\circ\text{C}$ 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 expression is 15 - 20 = 15 + (-20) = -5°C, as it rewrites the subtraction properly and interprets the temperature drop. A common error is rewriting as addition without the negative, like in choice A where 15 - 20 = 15 + 20 = 35°C denies the inverse, or misapplying the context as in choice D where reversing the subtraction gives a positive without the inverse. 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).
20. A science lab starts at $15^\circ\text{C}$ and then the temperature drops by $20^\circ\text{C}$. Which choice correctly models the situation, rewrites subtraction as addition, and gives the final temperature?
    Answer: $15-20=15+(-20)=-5^\circ\text{C}$ 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.

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