12 multiple-choice questions, progressively harder.
Evaluate ∣(−7)+2∣\lvert (-7) + 2 \rvert∣(−7)+2∣.
Solution
Correct answer: B
Add inside the bars first. Adding a positive to a negative moves right on the line, so starting at −7-7−7 and adding 222 lands at −5-5−5.
∣(−7)+2∣=∣−5∣=5\lvert (-7) + 2 \rvert = \lvert -5 \rvert = 5∣(−7)+2∣=∣−5∣=5
The distance from zero to −5-5−5 is 555.
What is the distance between 444 and 111111 on the number line?
Correct answer: C
The distance between two numbers is the absolute value of their difference.
∣4−11∣=∣−7∣=7\lvert 4 - 11 \rvert = \lvert -7 \rvert = 7∣4−11∣=∣−7∣=7
The two numbers are seven units apart.
Evaluate ∣8−8∣\lvert 8 - 8 \rvert∣8−8∣.
Correct answer: A
Work inside the bars first: 8−8=08 - 8 = 08−8=0. The absolute value of 000 is 000, since zero is no distance from itself.
∣8−8∣=∣0∣=0\lvert 8 - 8 \rvert = \lvert 0 \rvert = 0∣8−8∣=∣0∣=0
How many integers have an absolute value of 555?
An integer five units from zero can lie on either side: five units right is 555, five units left is −5-5−5.
∣5∣=5and∣−5∣=5\lvert 5 \rvert = 5 \quad\text{and}\quad \lvert -5 \rvert = 5∣5∣=5and∣−5∣=5
That is two integers, 555 and −5-5−5.
Which statement is true?
A number and its opposite are the same distance from zero, so they share one absolute value.
∣−6∣=6=∣6∣\lvert -6 \rvert = 6 = \lvert 6 \rvert∣−6∣=6=∣6∣
So the first statement is true; the others are false.
What is the distance between −3-3−3 and 555 on the number line?
Correct answer: D
Distance is the absolute value of the difference.
∣−3−5∣=∣−8∣=8\lvert -3 - 5 \rvert = \lvert -8 \rvert = 8∣−3−5∣=∣−8∣=8
From −3-3−3 it is three one-unit steps to zero and five more to 555, which is eight steps in all.
Evaluate ∣9∣−∣−4∣\lvert 9 \rvert - \lvert -4 \rvert∣9∣−∣−4∣.
Each number has its own bars, so take each absolute value first, then subtract.
∣9∣−∣−4∣=9−4=5\lvert 9 \rvert - \lvert -4 \rvert = 9 - 4 = 5∣9∣−∣−4∣=9−4=5
How many integers have an absolute value of 000?
An absolute value of 000 means a distance of 000 from zero, so the number must sit exactly on zero.
∣0∣=0\lvert 0 \rvert = 0∣0∣=0
Only 000 works, so there is exactly one such integer.
Evaluate ∣5−12∣\lvert 5 - 12 \rvert∣5−12∣.
Subtract inside the bars first: 5−12=−75 - 12 = -75−12=−7. Then take the absolute value of that result.
∣5−12∣=∣−7∣=7\lvert 5 - 12 \rvert = \lvert -7 \rvert = 7∣5−12∣=∣−7∣=7
Which number is farthest from zero: −12-12−12, 999, −7-7−7, or 101010?
Farthest from zero means the greatest absolute value.
∣−12∣=12,∣9∣=9,∣−7∣=7,∣10∣=10\lvert -12 \rvert = 12, \quad \lvert 9 \rvert = 9, \quad \lvert -7 \rvert = 7, \quad \lvert 10 \rvert = 10∣−12∣=12,∣9∣=9,∣−7∣=7,∣10∣=10
The largest distance is 121212, so −12-12−12 is farthest from zero.
What is the distance between −2-2−2 and −9-9−9 on the number line?
∣−2−(−9)∣=∣−2+9∣=∣7∣=7\lvert -2 - (-9) \rvert = \lvert -2 + 9 \rvert = \lvert 7 \rvert = 7∣−2−(−9)∣=∣−2+9∣=∣7∣=7
Subtracting a negative adds its opposite, giving 777 units between them.
Evaluate ∣0−6∣\lvert 0 - 6 \rvert∣0−6∣.
Work inside first: 0−6=−60 - 6 = -60−6=−6. Then take the absolute value of that result.
∣0−6∣=∣−6∣=6\lvert 0 - 6 \rvert = \lvert -6 \rvert = 6∣0−6∣=∣−6∣=6
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