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.

When Sarah writes a check, she subtracts that amount from her balance. Using additive inverse: $42.50 - $67.25 = $42.50 + (-$67.25) = -$24.75. This negative result indicates she is overdrawn by $24.75. Choice A adds instead of subtracts. Choice C uses double negative incorrectly. Choice D reverses the operation order.

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.

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.

When finding the distance between two points on a coordinate plane, you need to carefully examine their coordinates to determine the most efficient approach.

Looking at points A at $$(-3.2, 4.7)$$ and B at $$(1.8, 4.7)$$, notice that both points have the same y-coordinate: $$4.7$$. This means the points lie on the same horizontal line, so you only need to find the horizontal distance between them.

To find the horizontal distance, subtract the x-coordinates and take the absolute value: $$|(-3.2) - 1.8| = |-5.0| = 5.0$$ units. This matches answer choice D.

Let's examine why the other options are incorrect:

Answer A applies the distance formula correctly and gets the right numerical answer (5.0), but the question specifically asks you to use "the absolute value of the difference," not the distance formula. While mathematically valid, it doesn't follow the given instructions.

Answer B incorrectly adds the x-coordinates instead of subtracting them: $$|(-3.2) + 1.8|$$. This fundamental error in the distance calculation gives $$1.4$$ units instead of the correct $$5.0$$ units.

Answer C focuses on the y-coordinates, calculating $$|4.7 - 4.7| = 0$$, and incorrectly concludes the points are identical. While the y-coordinates are the same, the x-coordinates are different, so the points are definitely not the same.

Study tip: When points share the same coordinate (either x or y), you can find the distance by taking the absolute value of the difference of the other coordinate. Always subtract coordinates in the same order to avoid confusion.

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.

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 10-25 is 10+(-25)=-15, interpreting as short $15 coins (negative balance), as in choice A. A common error is misapplying signs in context, like choice B adding positive 25 to get 35 (positive balance instead of debt), or choice C with incorrect sum of -15 as 15, or choice D subtracting -25 which adds 25 incorrectly. 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); contexts like money $10 spend $25 (10-25=-15, debt). Mistakes include context misapplied (debt shown as positive balance), or not rewriting as addition (missing p-q=p+(-q)).

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).

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). The distance between -4 and 3 is |-4-3|=|-7|=7, matching choice A. A common error is choosing B (-4-3=-7), which gives a negative without absolute value, or C (|3-(-4)|=-7), which incorrectly assigns a negative to the absolute value. 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).

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 expression is |3-10|=|-7|=7, so distance 7, as in choice C, using absolute value properly. A common error is distance negative, like choice A |10-3|=-7 (wrong absolute), or choice B 10-3=7 without absolute value (works but not general), or choice D |10-3|=|13|=13 (arithmetic error). Distance: calculate p-q, take |p-q| (positive: 3-10=-7, |-7|=7), or |q-p| (same); mistakes include distance without absolute value (negative distance), or wrong order.