12 multiple-choice questions, progressively harder.
Which integer satisfies BOTH n<−2n < -2n<−2 and n>−6n > -6n>−6?
Solution
Correct answer: C
The two conditions place nnn strictly between −6-6−6 and −2-2−2, so the candidates are −5,−4,−3-5, -4, -3−5,−4,−3.
−6<−4<−2-6 < -4 < -2−6<−4<−2
Of the choices, only −4-4−4 falls in that range.
A number is 555 ticks to the left of −2-2−2 on the number line. What number is it?
Moving left makes a number smaller, so count down five ticks from −2-2−2.
−2→ 5 left −7-2 \xrightarrow{\,5\text{ left}\,} -7−25 left−7
So five ticks left of −2-2−2 lands on −7-7−7.
An account balance is −60-60−60 dollars. Which balance would show a LARGER debt?
Correct answer: B
A larger debt means owing more, which is a more negative balance, farther to the left on the line.
−80<−60-80 < -60−80<−60
So a balance of −80-80−80 dollars shows the larger debt.
City A records a low of −2∘-2^\circ−2∘. City B is colder than City A but warmer than −6∘-6^\circ−6∘. Which could be City B's low?
Correct answer: D
Colder than −2∘-2^\circ−2∘ means smaller than −2-2−2, and warmer than −6∘-6^\circ−6∘ means larger than −6-6−6, so City B lies strictly between −6-6−6 and −2-2−2.
Of the choices, only −4∘-4^\circ−4∘ fits both conditions.
Which of these numbers is farthest from zero?
Correct answer: A
Distance from zero is the number of units, regardless of sign. The distances here are 151515, 121212, 999, and 101010.
15>12>10>915 > 12 > 10 > 915>12>10>9
So −15-15−15 is farthest from zero.
A number's opposite is less than the number itself. What can you conclude about the number?
If a number's opposite is the smaller of the two, then the number sits to the right of its opposite. A number lies to the right of its opposite exactly when it is positive.
(opposite of n)<0<n(\text{opposite of } n) < 0 < n(opposite of n)<0<n
So the number must be positive.
On a number line, −7-7−7 and another point are the same distance from −4-4−4, one on each side. What is the other point?
The point −7-7−7 is three units to the left of −4-4−4, so the other point is three units to the right of −4-4−4.
−4→ 3 right −1-4 \xrightarrow{\,3\text{ right}\,} -1−43 right−1
So the other point is −1-1−1.
Which ordering from greatest to least is correct for −7, 2, −1, −12, 0-7,\; 2,\; -1,\; -12,\; 0−7,2,−1,−12,0?
Greatest to least reads right to left on the line, so the most positive comes first and the most negative comes last.
2>0>−1>−7>−122 > 0 > -1 > -7 > -122>0>−1>−7>−12
The opposite of a number nnn is greater than nnn itself. What can you conclude about nnn?
If the opposite of nnn is the larger of the two, then nnn sits to the left of its opposite on the line. A number lies to the left of its opposite exactly when it is negative.
n<0<(opposite of n)n < 0 < (\text{opposite of } n)n<0<(opposite of n)
So nnn must be negative.
Which integer lies strictly between −10-10−10 and −6-6−6 and is also farther from zero than −7-7−7?
The integers strictly between −10-10−10 and −6-6−6 are −9,−8,−7-9, -8, -7−9,−8,−7. Among these candidates, which are all negative, being farther from zero than −7-7−7 means lying to the left of −7-7−7, which rules out −7-7−7 itself.
−8<−7 and −10<−8<−6-8 < -7 \text{ and } -10 < -8 < -6−8<−7 and −10<−8<−6
That leaves −9-9−9 and −8-8−8, and only −8-8−8 is offered.
Which integer is NOT strictly between −9-9−9 and −1-1−1 on the number line?
Strictly between −9-9−9 and −1-1−1 means the values from −8-8−8 up to −2-2−2, with both endpoints left out. The number −10-10−10 lies even farther left than −9-9−9.
−10<−9-10 < -9−10<−9
So −10-10−10 is not between them.
Which statement correctly compares −2-2−2 and −20-20−20?
Comparison is decided by position on the line, not by the size of the digits. The point −2-2−2 is closer to zero than −20-20−20, so it sits farther right.
−2>−20-2 > -20−2>−20
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