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Distinguish Absolute Value From Order - 6th Grade Math

  1. A student compares −3-3 and −10-10. Which statement correctly compares them by absolute value and by order?
    Answer: ∣−3∣<∣−10∣|-3|<|-10| and −3>−10-3>-10 This question tests distinguishing absolute value comparisons (magnitudes: |-3| vs |-10| distances from zero) from order comparisons (values: -3 vs -10 which is greater on the number line). Absolute value compares magnitudes: |-3|=3 and |-10|=10, compare: 3<10 so |-3|<|-10| (-3 has smaller distance from zero, smaller magnitude); order compares values: -3 and -10 on number line, -3 right of -10 (less negative is greater), so -3>-10 (value comparison includes signs). For example, compare -3 and -10: |-3|=3<10=|-10| (-10 farther from zero), order -3>-10; or temperatures -3°C vs -10°C: |-3|=3<|-10|=10, order -3>-10 (-3 warmer); debt -$3 vs -$10: |-3|=3<|-10|=10, but -3>-10 (smaller debt). The correct distinction is that the magnitude of -3 is less than -10, while the value of -3 is greater than -10, so choice A is right. A common error is thinking order determines magnitude, like -3>-10 so |-3|>|-10|, but actually |-3|<|-10|; or reversing the order for negatives. To compare by absolute value: (1) calculate |a| and |b| (remove signs), (2) compare magnitudes (which farther from zero? |-3|=3<10=|-10|), (3) statement: |-3|<|-10|; for order: (1) locate on number line (which left/right?), (2) compare values (include signs: -10 left of -3), (3) statement: -3>-10. These can differ: among negatives, larger magnitude means smaller value.
  2. A student compares −16-16 and 99. Which choice correctly explains the difference between comparing absolute values and comparing the numbers themselves?
    Answer: Absolute value compares distance from 00, so ∣−16∣>∣9∣|-16|>|9| but −16<9-16<9 when comparing the actual values. This question tests distinguishing absolute value comparisons, which measure magnitudes as distances from zero on the number line, from order comparisons, which determine which value is greater by their positions on the number line. Absolute value compares magnitudes: |-16|=16 and |9|=9, so compare 16>9, meaning |-16|>|9| because -16 has a greater distance from zero and larger magnitude regardless of direction; order compares values: -16 and 9 on the number line, with -16 to the left of 9 (negative is less than positive), so -16<9, as value comparison includes signs. The distinction is that a number can have greater magnitude while having a lesser value, so |-16|>|9| is true for magnitudes, but -16<9 is true for values—these are different comparisons. For example, compare -16 and 9: absolute values |-16|=16 and |9|=9, magnitude 16>9 so |-16|>|9| (-16 is farther from zero), order -16<9 (negative is less than positive). The correct distinction is that choice B explains absolute value as distance from 0, so |-16|>|9| but -16<9 for actual values, correctly separating the concepts unlike others. A common error is conflating them, like saying absolute value compares which is greater on the number line (as in A, wrongly implying -16>9), or keeping the negative sign in absolute value (as in C, |-16|≠-16), or ignoring signs in order (as in D). To compare absolute values: (1) calculate |a| and |b| by removing signs, (2) compare which is farther from zero, like 16>9 so |-16|>|9|; for order: (1) locate on the number line, (2) compare values including signs, -16 is left of 9 so -16<9—these differ because a negative number can have large magnitude but small value, like |-100|>|1| but -100<1; mistakes include not recognizing they can differ or applying wrong in context.
  3. On a number line, a diver is −9-9 meters (below sea level) and a bird is 44 meters (above sea level). Which statement is true about their distances from sea level and their positions?
    Answer: ∣−9∣>∣4∣|-9|>|4| and −9<4-9<4 This question tests distinguishing absolute value comparisons, which measure magnitudes as distances from zero on the number line, from order comparisons, which determine which value is greater by their positions on the number line. Absolute value compares magnitudes: |-9|=9 and |4|=4, so compare 9>4, meaning |-9|>|4| because -9 has a greater distance from zero and larger magnitude regardless of direction; order compares values: -9 and 4 on the number line, with -9 to the left of 4 (negative is less than positive), so -9<4, as value comparison includes signs. The distinction is that a number can have greater magnitude while having a lesser value, so |-9|>|4| is true for magnitudes, but -9<4 is true for values—these are different comparisons, like in elevations where -9m (below sea level) is farther from zero (greater magnitude) but lower position (smaller value) than 4m above. For example, positions -9m vs 4m: absolute values |-9|=9 and |4|=4, magnitude 9>4 so |-9|>|4| (-9 is farther from sea level), order -9<4 (below is less than above). The correct distinction is that choice B accurately states the magnitude comparison (|-9|>|4|) and the value comparison (-9<4), unlike others that misstate one or both. A common error is thinking magnitude determines order, like claiming |-9|>|4| implies -9>4 (as in C, but actually -9<4), or wrongly calculating absolute values like assuming |-9|<|4| (as in A) or equality (as in D). To compare absolute values: (1) calculate |a| and |b| by removing signs, (2) compare which is farther from zero, like 9>4 so |-9|>|4|; for order: (1) locate on the number line, (2) compare values including signs, -9 is left of 4 so -9<4—these differ because a negative number can have large magnitude but small value, like |-100|>|1| but -100<1; in elevations, use magnitude for distance from sea level, value for which is higher.
  4. A submarine is at −18-18 meters relative to sea level, and a bird is at 22 meters relative to sea level. Which statement correctly compares their positions by absolute value and by order?
    Answer: ∣−18∣>∣2∣|-18|>|2| and −18<2-18<2 This question tests distinguishing absolute value comparisons (magnitudes: |-18| vs |2| distances from zero) from order comparisons (values: -18 vs 2 which is greater on the number line). Absolute value compares magnitudes: |-18|=18 and |2|=2, compare: 18>2 so |-18|>|2| (-18 has greater distance from zero, larger magnitude regardless of direction). Order compares values: -18 and 2 on number line, -18 left of 2 (negative < positive), so -18<2 (value comparison includes signs). Distinction: magnitude can be greater while value is less (|-18|>|2| magnitude comparison true, but -18<2 value comparison also true—different comparisons). In submarine and bird context: |-18|=18>|2|=2 (submarine farther from sea level), but -18<2 (below is less than above). The correct statement is |-18|>|2| and -18<2, separating depth magnitude from elevation order. Common mistakes: thinking magnitudes are equal or that larger magnitude implies larger order.
  5. Sarah says: "If x>y>0x > y > 0, then ∣x∣>∣y∣|x| > |y|." Is Sarah correct?
    Answer: Yes, when both numbers are positive, ∣x∣=x|x| = x and ∣y∣=y|y| = y Sarah is correct. When both numbers are positive, the absolute value equals the number itself: ∣x∣=x|x| = x and ∣y∣=y|y| = y. So x>yx > y means ∣x∣>∣y∣|x| > |y|. This is a special case where order and absolute value comparisons give the same result.
  6. A game score change is −12-12 points (lost points) and another change is 77 points (gained points). Which statement is true about the magnitude and the order of the numbers −12-12 and 77?
    Answer: ∣−12∣>∣7∣|-12|>|7| and −12<7-12<7 This question tests distinguishing absolute value comparisons (magnitudes: |-12| vs |7| distances from zero) from order comparisons (values: -12 vs 7 which is greater on the number line). Absolute value compares magnitudes: |-12|=12 and |7|=7, compare: 12>7 so |-12|>|7| (-12 has greater distance from zero, larger magnitude regardless of direction); order compares values: -12 and 7 on number line, -12 left of 7 (negative < positive), so -12<7 (value comparison includes signs). For example, in game scores: |-12|=12 vs |7|=7, magnitude 12>7 (-12 is a larger point change), order -12<7 (loss is less than gain); or temperature: |-15|=15>|10|=10, order -15<10; debt -$50 vs $30: |-50|=50>|30|=30, but -50<30. The correct distinction is that the magnitude of -12 is greater than that of 7 (bigger change), while the value of -12 is less than 7, so choice C is right. A common error is confusing magnitude with order, like thinking |-12|>|7| implies -12>7, but actually -12<7; or reversing the magnitude comparison; or treating them as equal. To compare by absolute value: (1) calculate |a| and |b| (remove signs), (2) compare magnitudes (which farther from zero? |-12|=12>7=|7|), (3) statement: |-12|>|7|; for order: (1) locate on number line (which left/right?), (2) compare values (include signs: -12 left of 7), (3) statement: -12<7. These can differ: large negative changes have big magnitude but small value; in scores, magnitude for size of change, order for net effect.
  7. A student compares the integers −8-8 and 55 in two ways: by absolute value (magnitude) and by order (which number is greater). Which statement is true?
    Answer: ∣−8∣>∣5∣|-8|>|5| and −8<5-8<5 This question tests distinguishing absolute value comparisons, which measure magnitudes as distances from zero on the number line, from order comparisons, which determine which value is greater by their positions on the number line. Absolute value compares magnitudes: |-8|=8 and |5|=5, so compare 8>5, meaning |-8|>|5| because -8 has a greater distance from zero and larger magnitude regardless of direction; order compares values: -8 and 5 on the number line, with -8 to the left of 5 (negative is less than positive), so -8<5, as value comparison includes signs. The distinction is that a number can have greater magnitude while having a lesser value, so |-8|>|5| is true for magnitudes, but -8<5 is true for values—these are different comparisons. For example, compare -8 and 5: absolute values |-8|=8 and |5|=5, magnitude 8>5 so |-8|>|5| (-8 is farther from zero), order -8<5 (negative is less than positive on the number line). The correct distinction is that choice B accurately states the magnitude comparison (|-8|>|5|) and the value comparison (-8<5), unlike others that misstate one or both. A common error is thinking magnitude determines order, like claiming |-8|>|5| implies -8>5 (as in C, but actually -8<5), or wrongly calculating absolute values like assuming |-8|<|5| (as in A) or equality (as in D). To compare absolute values: (1) calculate |a| and |b| by removing signs, (2) compare which is farther from zero, like 8>5 so |-8|>|5|; for order: (1) locate on the number line, (2) compare values including signs, -8 is left of 5 so -8<5—these differ because a negative number can have large magnitude but small value, like |-100|>|1| but -100<1.
  8. During a science lab, the temperature in a freezer is −15∘C-15^\circ\text{C} and the temperature in a room is 10∘C10^\circ\text{C}. Which statement correctly compares the temperatures by absolute value (distance from 0∘C0^\circ\text{C}) and by order (which temperature is greater)?
    Answer: ∣−15∣>∣10∣|-15|>|10| and −15<10-15<10 This question tests distinguishing absolute value comparisons (magnitudes: |-15| vs |10| distances from zero) from order comparisons (values: -15 vs 10 which is greater on the number line). Absolute value compares magnitudes: |-15|=15 and |10|=10, compare: 15>10 so |-15|>|10| (-15 has greater distance from zero, larger magnitude regardless of direction); order compares values: -15 and 10 on number line, -15 left of 10 (negative < positive), so -15<10 (value comparison includes signs). For example, in temperature context: |-15|=15 vs |10|=10, magnitude 15>10 (-15 more extreme from 0°C, greater magnitude deviation), order -15<10 (-15 colder, smaller value); or compare -8 and 5: |-8|=8>5=|5| (-8 farther from zero), order -8<5; debt -$50 vs $30: |-50|=50>|30|=30, but -50<30. The correct distinction is that the magnitude of -15°C is greater than that of 10°C (farther from 0°C), while the value of -15°C is less than 10°C (colder), so choice A is right. A common error is using magnitude for order, like claiming |-15|>|10| means -15>10, but actually -15<10; or wrongly thinking magnitudes are equal or reversed; or misapplying context, like saying -15 is warmer because of larger magnitude. To compare by absolute value: (1) calculate |a| and |b| (remove signs), (2) compare magnitudes (which farther from zero? |-15|=15>10=|10|), (3) statement: |-15|>|10|; for order: (1) locate on number line (which left/right?), (2) compare values (include signs: -15 left of 10), (3) statement: -15<10. These can differ: negative numbers can have large magnitude but small value; in temperature, use magnitude for deviation from zero, order for which is warmer.
  9. A student compares the integers −8-8 and 55 in two ways: by absolute value (magnitude) and by order (which number is greater). Which statement is true?​​
    Answer: ∣−8∣>∣5∣|-8|>|5| and −8<5-8<5 This question tests distinguishing absolute value comparisons, which measure magnitudes as distances from zero on the number line, from order comparisons, which determine which value is greater by their positions on the number line. Absolute value compares magnitudes: |-8|=8 and |5|=5, so compare 8>5, meaning |-8|>|5| because -8 has a greater distance from zero and larger magnitude regardless of direction; order compares values: -8 and 5 on the number line, with -8 to the left of 5 (negative is less than positive), so -8<5, as value comparison includes signs. The distinction is that a number can have greater magnitude while having a lesser value, so |-8|>|5| is true for magnitudes, but -8<5 is true for values—these are different comparisons. For example, compare -8 and 5: absolute values |-8|=8 and |5|=5, magnitude 8>5 so |-8|>|5| (-8 is farther from zero), order -8<5 (negative is less than positive on the number line). The correct distinction is that choice B accurately states the magnitude comparison (|-8|>|5|) and the value comparison (-8<5), unlike others that misstate one or both. A common error is thinking magnitude determines order, like claiming |-8|>|5| implies -8>5 (as in C, but actually -8<5), or wrongly calculating absolute values like assuming |-8|<|5| (as in A) or equality (as in D). To compare absolute values: (1) calculate |a| and |b| by removing signs, (2) compare which is farther from zero, like 8>5 so |-8|>|5|; for order: (1) locate on the number line, (2) compare values including signs, -8 is left of 5 so -8<5—these differ because a negative number can have large magnitude but small value, like |-100|>|1| but -100<1.
  10. Give two numbers where the smaller number has the larger absolute value.
    Answer: −8-8 and −2-2: −8<−2-8 < -2 and ∣−8∣=8>2|-8| = 8 > 2 In option B, −8-8 is smaller than −2-2 (since −8<−2-8 < -2), but −8-8 has the larger absolute value since ∣−8∣=8>2=∣−2∣|-8| = 8 > 2 = |-2|. This shows how the smaller number can have the larger absolute value.
  11. A hiker's elevation change is −14-14 meters (downhill) and a biker's elevation change is 66 meters (uphill). Which statement correctly compares the sizes of the changes and the actual changes?
    Answer: ∣−14∣>∣6∣|-14|>|6| and −14<6-14<6 This question asks you to compare two things: the size of each change and the actual change. The size of a change is its absolute value — how far it is from zero, ignoring the direction. The hiker's change has size ∣−14∣=14|-14| = 14, and the biker's change has size ∣6∣=6|6| = 6. Since 1414 is bigger than 66, we write ∣−14∣>∣6∣|-14| > |6|. The actual change keeps the sign. On a number line, −14-14 sits to the left of 66, so −14<6-14 < 6. (Negative numbers are always less than positive ones!) Putting both together gives us choice B. Think of it like elevators: the hiker dropped 1414 floors, the biker rose 66. The hiker moved farther (14>614 > 6), but ended up lower (−14<6-14 < 6). Try this at home: pick two temperatures, like −9°-9° and 4°4°. Compare their sizes, then compare the actual values!
  12. In science class, the temperature in a freezer is −15∘C-15^\circ\text{C} and the temperature in a classroom is 10∘C10^\circ\text{C}. Which comparison correctly describes both the magnitudes and the actual temperatures?​​
    Answer: ∣−15∣>∣10∣|-15|>|10| and −15<10-15<10 This question tests distinguishing absolute value comparisons, which measure magnitudes as distances from zero on the number line, from order comparisons, which determine which value is greater by their positions on the number line. Absolute value compares magnitudes: |-15|=15 and |10|=10, so compare 15>10, meaning |-15|>|10| because -15 has a greater distance from zero and larger magnitude regardless of direction; order compares values: -15 and 10 on the number line, with -15 to the left of 10 (negative is less than positive), so -15<10, as value comparison includes signs. The distinction is that a number can have greater magnitude while having a lesser value, so |-15|>|10| is true for magnitudes, but -15<10 is true for values—these are different comparisons, like in temperature context where -15°C is more extreme from 0°C (greater magnitude deviation) but colder (smaller value) than 10°C. For example, temperature -15°C vs 10°C: absolute values |-15|=15 and |10|=10, magnitude 15>10 so |-15|>|10| (-15 is more extreme), order -15<10 (-15 is colder, smaller value on the scale). The correct distinction is that choice A accurately states the magnitude comparison (|-15|>|10|) and the value comparison (-15<10), unlike others that misstate one or both. A common error is thinking magnitude determines order, like claiming |-15|>|10| implies -15>10 (as in C, but actually -15<10), or wrongly calculating absolute values like assuming |-15|<|10| (as in B) or equality (as in D). To compare absolute values: (1) calculate |a| and |b| by removing signs, (2) compare which is farther from zero, like 15>10 so |-15|>|10|; for order: (1) locate on the number line, (2) compare values including signs, -15 is left of 10 so -15<10—these differ because a negative number can have large magnitude but small value, like |-100|>|1| but -100<1; in temperature, use magnitude for deviation from zero, value for which is warmer.
  13. A student compares −16-16 and 99. Which choice correctly explains the difference between comparing absolute values and comparing the numbers themselves?​​
    Answer: Absolute value compares distance from 00, so ∣−16∣>∣9∣|-16|>|9| but −16<9-16<9 when comparing the actual values. This question tests distinguishing absolute value comparisons, which measure magnitudes as distances from zero on the number line, from order comparisons, which determine which value is greater by their positions on the number line. Absolute value compares magnitudes: |-16|=16 and |9|=9, so compare 16>9, meaning |-16|>|9| because -16 has a greater distance from zero and larger magnitude regardless of direction; order compares values: -16 and 9 on the number line, with -16 to the left of 9 (negative is less than positive), so -16<9, as value comparison includes signs. The distinction is that a number can have greater magnitude while having a lesser value, so |-16|>|9| is true for magnitudes, but -16<9 is true for values—these are different comparisons. For example, compare -16 and 9: absolute values |-16|=16 and |9|=9, magnitude 16>9 so |-16|>|9| (-16 is farther from zero), order -16<9 (negative is less than positive). The correct distinction is that choice B explains absolute value as distance from 0, so |-16|>|9| but -16<9 for actual values, correctly separating the concepts unlike others. A common error is conflating them, like saying absolute value compares which is greater on the number line (as in A, wrongly implying -16>9), or keeping the negative sign in absolute value (as in C, |-16|≠-16), or ignoring signs in order (as in D). To compare absolute values: (1) calculate |a| and |b| by removing signs, (2) compare which is farther from zero, like 16>9 so |-16|>|9|; for order: (1) locate on the number line, (2) compare values including signs, -16 is left of 9 so -16<9—these differ because a negative number can have large magnitude but small value, like |-100|>|1| but -100<1; mistakes include not recognizing they can differ or applying wrong in context.
  14. A bank app shows two balances: Account 1 is −11-11 dollars (owed) and Account 2 is 33 dollars (saved). Which statement correctly compares the amounts by magnitude and the balances by value?
    Answer: ∣−11∣>∣3∣|-11|>|3| and −11<3-11<3 This question tests distinguishing absolute value comparisons, which measure magnitudes as distances from zero on the number line, from order comparisons, which determine which value is greater by their positions on the number line. Absolute value compares magnitudes: |-11|=11 and |3|=3, so compare 11>3, meaning |-11|>|3| because -11 has a greater distance from zero and larger magnitude regardless of direction; order compares values: -11 and 3 on the number line, with -11 to the left of 3 (negative is less than positive), so -11<3, as value comparison includes signs. The distinction is that a number can have greater magnitude while having a lesser value, so |-11|>|3| is true for magnitudes, but -11<3 is true for values—these are different comparisons, like in bank balances where -$11 is a larger amount owed (greater magnitude) but lower balance (smaller value) than $3 saved. For example, balances -11 vs 3: absolute values |-11|=11 and |3|=3, magnitude 11>3 so |-11|>|3| (larger amount), order -11<3 (owed is less than saved). The correct distinction is that choice A accurately states the magnitude comparison (|-11|>|3|) and the value comparison (-11<3), unlike others that misstate one or both. A common error is thinking magnitude determines order, like claiming |-11|>|3| implies -11>3 (as in C, but actually -11<3), or wrongly calculating absolute values like assuming |-11|<|3| (as in B) or equality (as in D). To compare absolute values: (1) calculate |a| and |b| by removing signs, (2) compare which is farther from zero, like 11>3 so |-11|>|3|; for order: (1) locate on the number line, (2) compare values including signs, -11 is left of 3 so -11<3—these differ because a negative number can have large magnitude but small value, like |-100|>|1| but -100<1; in banking, use magnitude for debt size, value for net worth.
  15. Which pair of statements is true for the integers −10-10 and 33 (one statement about absolute value and one about order)?​​
    Answer: ∣−10∣>∣3∣|-10|>|3| and −10<3-10<3 This question tests distinguishing absolute value comparisons, which measure magnitudes as distances from zero on the number line, from order comparisons, which determine which value is greater by their positions on the number line. Absolute value compares magnitudes: |-10|=10 and |3|=3, so compare 10>3, meaning |-10|>|3| because -10 has a greater distance from zero and larger magnitude regardless of direction; order compares values: -10 and 3 on the number line, with -10 to the left of 3 (negative is less than positive), so -10<3, as value comparison includes signs. The distinction is that a number can have greater magnitude while having a lesser value, so |-10|>|3| is true for magnitudes, but -10<3 is true for values—these are different comparisons. For example, compare -10 and 3: absolute values |-10|=10 and |3|=3, magnitude 10>3 so |-10|>|3| (-10 is farther from zero), order -10<3 (negative is less than positive). The correct distinction is that choice C accurately states the magnitude comparison (|-10|>|3|) and the value comparison (-10<3), unlike others that misstate one or both. A common error is thinking magnitude determines order, like claiming |-10|>|3| implies -10>3 (as in B, but actually -10<3), or wrongly calculating absolute values like assuming |-10|<|3| (as in A) or equality (as in D). To compare absolute values: (1) calculate |a| and |b| by removing signs, (2) compare which is farther from zero, like 10>3 so |-10|>|3|; for order: (1) locate on the number line, (2) compare values including signs, -10 is left of 3 so -10<3—these differ because a negative number can have large magnitude but small value, like |-100|>|1| but -100<1.
  16. Compare −7-7 and 77 in two ways. Which statement is true?​​
    Answer: ∣−7∣=∣7∣|-7|=|7| and −7<7-7<7 This question tests distinguishing absolute value comparisons, which measure magnitudes as distances from zero on the number line, from order comparisons, which determine which value is greater by their positions on the number line. Absolute value compares magnitudes: |-7|=7 and |7|=7, so compare 7=7, meaning |-7|=|7| because both have the same distance from zero and equal magnitude regardless of direction; order compares values: -7 and 7 on the number line, with -7 to the left of 7 (negative is less than positive), so -7<7, as value comparison includes signs. The distinction is that numbers can have equal magnitude but different values, so |-7|=|7| is true for magnitudes, but -7<7 is true for values—these are different comparisons. For example, compare -7 and 7: absolute values |-7|=7 and |7|=7, magnitude 7=7 so |-7|=|7| (same distance), order -7<7 (negative is less than positive). The correct distinction is that choice B accurately states the magnitude comparison (|-7|=|7|) and the value comparison (-7<7), unlike others that misstate one or both. A common error is thinking equal magnitude implies equal or reversed order, like claiming -7>7 (as in C), or wrongly calculating absolute values like assuming |-7|>|7| (as in A) or |-7|<|7| (as in D). To compare absolute values: (1) calculate |a| and |b| by removing signs, (2) compare which is farther from zero, like 7=7 so |-7|=|7|; for order: (1) locate on the number line, (2) compare values including signs, -7 is left of 7 so -7<7—these differ because opposites have equal magnitude but the negative has smaller value; use magnitude for symmetry questions, value for inequality.
  17. A student says, "Because ∣−6∣>∣2∣|-6|>|2|, it must be true that −6>2-6>2." Which choice correctly evaluates the student's claim using both absolute value and order comparisons?
    Answer: The student is incorrect: ∣−6∣>∣2∣|-6|>|2| but −6<2-6<2. This question tests distinguishing absolute value comparisons (magnitudes: |-6| vs |2| distances from zero) from order comparisons (values: -6 vs 2 which is greater on the number line). Absolute value compares magnitudes: |-6|=6 and |2|=2, compare: 6>2 so |-6|>|2| (-6 has greater distance from zero, larger magnitude regardless of direction); order compares values: -6 and 2 on number line, -6 left of 2 (negative < positive), so -6<2 (value comparison includes signs). For example, compare -6 and 2: |-6|=6>2=|2| (-6 farther from zero), order -6<2; or temperature: |-15|=15>|10|=10, order -15<10; debt -$50 vs $30: |-50|=50>|30|=30, but -50<30. The correct distinction is that while the magnitude of -6 is greater than 2, the value of -6 is less than 2, showing the student's claim is incorrect, so choice B is right. The student's error is assuming magnitude determines order (|-6|>|2| so -6>2 claimed, but -6<2); other errors include wrong magnitude comparison or thinking they are equal. To compare by absolute value: (1) calculate |a| and |b| (remove signs), (2) compare magnitudes (which farther from zero? |-6|=6>2=|2|), (3) statement: |-6|>|2|; for order: (1) locate on number line (which left/right?), (2) compare values (include signs: -6 left of 2), (3) statement: -6<2. These can differ: magnitude ignores sign, order considers it; mistakes conflate them.
  18. A student writes two statements about the integers −16-16 and 1212: 1) ∣−16∣>∣12∣|-16|>|12| 2) −16>12-16>12 Which choice correctly tells whether each statement is true or false?
    Answer: Statement 1 is true, and Statement 2 is false. This question tests distinguishing absolute value comparisons (magnitudes: |-16| vs |12| distances from zero) from order comparisons (values: -16 vs 12 which is greater on the number line). Absolute value compares magnitudes: |-16|=16 and |12|=12, compare: 16>12 so |-16|>|12| (-16 has greater distance from zero, larger magnitude regardless of direction); order compares values: -16 and 12 on number line, -16 left of 12 (negative < positive), so -16<12 (value comparison includes signs). For example, compare -16 and 12: |-16|=16>12=|12| (-16 farther from zero), order -16<12; or temperature: |-15|=15>|10|=10, order -15<10; debt -$50 vs $30: |-50|=50>|30|=30, but -50<30. The correct evaluation is that Statement 1 is true (magnitude comparison) and Statement 2 is false (order is actually -16<12), so choice C is right. A common error is conflating the two, like thinking |-16|>|12| makes -16>12 true, or wrongly assessing the magnitude as false. To compare by absolute value: (1) calculate |a| and |b| (remove signs), (2) compare magnitudes (which farther from zero? |-16|=16>12=|12|), (3) statement: |-16|>|12|; for order: (1) locate on number line (which left/right?), (2) compare values (include signs: -16 left of 12), (3) statement: -16<12. These can differ: Statement 2 mistakes magnitude for order.
  19. A game uses a score that can be negative. Player A has −12-12 points and Player B has 77 points. Which statement correctly compares their scores by absolute value and by order?​​
    Answer: ∣−12∣>∣7∣|-12|>|7| and −12<7-12<7 This question tests distinguishing absolute value comparisons, which measure magnitudes as distances from zero on the number line, from order comparisons, which determine which value is greater by their positions on the number line. Absolute value compares magnitudes: |-12|=12 and |7|=7, so compare 12>7, meaning |-12|>|7| because -12 has a greater distance from zero and larger magnitude regardless of direction; order compares values: -12 and 7 on the number line, with -12 to the left of 7 (negative is less than positive), so -12<7, as value comparison includes signs. The distinction is that a number can have greater magnitude while having a lesser value, so |-12|>|7| is true for magnitudes, but -12<7 is true for values—these are different comparisons, like in game scores where -12 points is a larger deficit (greater magnitude) but lower score (smaller value) than 7 points. For example, scores -12 vs 7: absolute values |-12|=12 and |7|=7, magnitude 12>7 so |-12|>|7| (-12 is larger in absolute terms), order -12<7 (negative is less than positive). The correct distinction is that choice C accurately states the magnitude comparison (|-12|>|7|) and the value comparison (-12<7), unlike others that misstate one or both. A common error is thinking magnitude determines order, like claiming |-12|>|7| implies -12>7 (as in B, but actually -12<7), or wrongly calculating absolute values like assuming |-12|<|7| (as in A) or equality (as in D). To compare absolute values: (1) calculate |a| and |b| by removing signs, (2) compare which is farther from zero, like 12>7 so |-12|>|7|; for order: (1) locate on the number line, (2) compare values including signs, -12 is left of 7 so -12<7—these differ because a negative number can have large magnitude but small value, like |-100|>|1| but -100<1; in scores, use magnitude for size of points, value for who is winning.
  20. A student writes: "Since ∣−15∣=15|-15| = 15 and ∣8∣=8|8| = 8, and 15>815 > 8, this means −15>8-15 > 8." What error did the student make?
    Answer: The student thought absolute value order equals number order The student correctly calculated the absolute values and compared them. However, the error was thinking that absolute value comparison determines the order of the original numbers. Absolute value measures distance from zero, not position on the number line. While ∣−15∣>∣8∣|-15| > |8|, we still have −15<8-15 < 8.

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