12 multiple-choice questions, progressively harder.
Which integer lies exactly halfway between −8-8−8 and −2-2−2 on the number line?
Solution
Correct answer: A
The halfway point sits the same number of ticks from each end. From −8-8−8 count right and from −2-2−2 count left until they meet.
−8→ 3 −5← 3 −2-8 \xrightarrow{\,3\,} -5 \xleftarrow{\,3\,} -2−83−53−2
So −5-5−5 is the midpoint, three units from each end.
How many integers lie strictly between −4-4−4 and 333 (not counting −4-4−4 or 333)?
Correct answer: C
List the integers from −3-3−3 up to 222, leaving out both endpoints.
−3, −2, −1, 0, 1, 2-3,\, -2,\, -1,\, 0,\, 1,\, 2−3,−2,−1,0,1,2
That is 666 integers.
Which statement is true for every negative integer nnn?
Correct answer: D
A negative integer lies to the left of zero, while its opposite lies the same distance to the right of zero. The left point is the smaller one.
n<0<(opposite of n)n < 0 < (\text{opposite of } n)n<0<(opposite of n)
So a negative integer is always less than its opposite.
Order from least to greatest: the opposite of 555, the number −2-2−2, and the opposite of −3-3−3.
The opposite of 555 is −5-5−5, and the opposite of −3-3−3 is 333, so the three values are −5-5−5, −2-2−2, and 333. Read them left to right on the line.
−5<−2<3-5 < -2 < 3−5<−2<3
Overnight, each temperature reading is lower than the one before it. Which list could record the readings in order from earliest to latest?
Lower means smaller, which is farther left on the line, so each reading must sit to the left of the one before it.
−1>−4>−9-1 > -4 > -9−1>−4>−9
So the list −1∘,−4∘,−9∘-1^\circ, -4^\circ, -9^\circ−1∘,−4∘,−9∘ drops at every step.
Which of these comparisons is FALSE?
Check each by position. The point −7-7−7 lies farther left than −6-6−6, so −7-7−7 is the smaller number.
−7<−6-7 < -6−7<−6
That makes "−7>−6-7 > -6−7>−6" the false statement; the other three are all true.
Point AAA is at −3-3−3 and point BBB is the opposite of AAA. Which point lies between AAA and BBB?
Correct answer: B
The opposite of −3-3−3 is 333, so B=3B = 3B=3. A point lies between AAA and BBB when it sits to the right of −3-3−3 and to the left of 333, so test the four choices by position.
−3<0<3-3 < 0 < 3−3<0<3
Only 000 passes both tests. The points −4-4−4 and −5-5−5 lie to the left of AAA, and 555 lies to the right of BBB.
If aaa is a negative integer and bbb is a positive integer, which is always true?
Every negative number lies to the left of zero and every positive number lies to the right of zero, so the negative is always farther left.
a<0<ba < 0 < ba<0<b
So a<ba < ba<b no matter their sizes.
Which ordering from greatest to least is correct: −3, 0, −10, −1, 4-3,\; 0,\; -10,\; -1,\; 4−3,0,−10,−1,4?
Greatest to least reads right to left on the line, so the most positive comes first and the most negative comes last.
4>0>−1>−3>−104 > 0 > -1 > -3 > -104>0>−1>−3>−10
Two cities report low temperatures. City A is −3∘-3^\circ−3∘, and City B is colder than City A but warmer than −7∘-7^\circ−7∘. Which could be City B's temperature?
Colder than −3∘-3^\circ−3∘ means smaller than −3-3−3, and warmer than −7∘-7^\circ−7∘ means larger than −7-7−7, so City B lies strictly between −7-7−7 and −3-3−3.
−7<−5<−3-7 < -5 < -3−7<−5<−3
Of the choices, only −5∘-5^\circ−5∘ fits both conditions.
How many integers sit at distance exactly 444 from zero?
Distance 444 from zero is reached four ticks to the left and four ticks to the right, giving one integer on each side.
−4 and 4-4 \text{ and } 4−4 and 4
So exactly 222 integers sit four units from zero.
Two divers descend from the surface. Diver A ends at −26-26−26 meters and Diver B ends at −24-24−24 meters. Which statement is correct?
On the number line, the point farther to the left (more negative) is the smaller number, and underwater that means deeper. Compare −26-26−26 and −24-24−24: because −26-26−26 sits farther from zero on the negative side, it lies to the left of −24-24−24.
−26<−24-26 < -24−26<−24
So Diver A, at −26-26−26 meters, is deeper than Diver B at −24-24−24 meters. Comparing 262626 and 242424 without their signs gives the wrong order.
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