Chapter 2 Integers And Rational Numbers
Section 2.3: Absolute Values of Rational Numbers
Page 77 Exercise 1 Answer
For a bank account, a positive balance means you deposited more money than you spent.
A balance of $0 means you deposited the same amount that you spent.
A negative balance means you spent more than you deposited. That is, you owe the bank money.
An account balance that represents an amount owed greater than $40 is then a negative balance with an absolute value greater than 40. A possible balance could then be −$60.00.
Result
Possible answer: −$60.00
Any balance that is negative and has an absolute value greater than 40 is a correct answer.
Read And Learn More: enVisionmath 2.0 Grade 6 Volume 1 Solutions
Page 77 Exercise 1 Answer
For a bank account, a positive balance means you deposited more money than you spent.
A balance of $0 means you deposited the same amount that you spent.
A negative balance means you spent more than you deposited. That is, you owe the bank money.
An ending balance of −$30.00 means you owe the bank $30.
Result
An ending balance of −$30.00 means you owe the bank $30.
Page 78 Exercise 1 Answer
The absolute value of a number is its distance from 0, which is always positive. The absolute value of a positive number is then the positive number and the absolute value of a negative number is its opposite.
The absolute values for each week are then:
Week 1: |-\(7 \frac{1}{2}\)| = \(7 \frac{1}{2}\)
Week 2: |2.2| = 2.2
Week 3: ∣−4.38∣ = 4.38
Since \(7 \frac{1}{2}\) is the largest absolute value, then the water level changed by the greatest amount in Week 1.
Yes, a lesser number can represent a greater change than a greater number because the lesser number could have a larger absolute value. In this problem, we saw this because –\(7 \frac{1}{2}\) was less than 2.2 and −4.38 but it represented the greatest change since it had the largest absolute value.
Result
Week 1: \(7 \frac{1}{2}\)
Week 2: 2.2
Week 3: 4.38
The water level changed by the greatest amount in Week 1.
Yes, a lesser number can represent a greater change than a greater number because the lesser number could have a larger absolute value.
Page 79 Exercise 2 Answer
Comparing -$19.45 and -$23.76, we know that -19.45 lies to the right of -20 on a number line while -23.76 lies to the left of -20. This means that -$23.76 must lie to the left of -$19.45 on a number line. Therefore, -$19.45 is the greater number.
To determine which balance is the lesser amount owed, we can compare their absolute values:
|-$19.45| = $19.45
|-$23.76| = $23.46
Since $19.45 < $23.76, then the balance -$19.45 is the lesser amount owed.
Result
V.Wong’s balance of -$19.45 is the greater number and the lesser amount owed.
Page 80 Exercise 1 Answer
Absolute values are used to describe a number’s distance from 0 on a number line.
Page 80 Exercise 2 Answer
The absolute value of a number is its distance form 0 on a number line.
Since −7 has a distance of 7 from 0 and 6 has a distance of 6 from 0, then −7 has a greater absolute value.
Result
Since −7 has a distance of 7 from 0 and 6 has a distance of 6 from 0, then −7 has a greater absolute value.
Page 80 Exercise 3 Answer
A balance that has a greater integer value than a balance of −$12 must be a balance that is greater than −12.
A debt is represented by a negative balance so if the balance also represents a debt of less than $5, then it must be a negative number and it must be bigger than −5.
A possible balance could then be −$3 since −3 > −12 and −$3 represents a debt of $3, which is less than $5.
Result
Possible answer: −$3
Any negative balance that is bigger than −5 is a correct answer.
Page 80 Exercise 4 Answer
We are given the three elevations of -2 feet, -12 feet, and 30 feet.
To determine which represents the least number, we need to think about which number is farthest left on a number line. The numbers on a number line would be in the order -12, -2, 30 so the least number is -12 feet.
To determine which elevation represents the farthest distance from sea level, we need to compare their absolute values:
|-2| = 2
|-12| = 12
|30| = 30
Since 30 has the largest absolute value, then 30 feet represents the farthest distance from sea level.
Result
−12 feet is the least number and 30 feet is the farthest distance from sea level.
Page 80 Exercise 5 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
∣−9∣ = 9
Result
9
Page 80 Exercise 6 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|\(5 \frac{3}{4}\)| = \(5 \frac{3}{4}\)
Result
\(5 \frac{3}{4}\)Page 80 Exercise 7 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
∣−5.5∣ = 5.5
Result
5.5
Page 80 Exercise 8 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|82.5| = 82.5
Result
82.5
Page 80 Exercise 9 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|-\(14 \frac{1}{3}\)| = \(14 \frac{1}{3}\)
Result
\(14 \frac{1}{3}\)Page 80 Exercise 10 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
∣−7.75∣ = 7.75
Result
7.75
Page 80 Exercise 11 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
∣−19∣ = 19
Result
19
Page 80 Exercise 12 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|-\(2 \frac{1}{2}\)| = \(2 \frac{1}{2}\)
Result
\(2 \frac{1}{2}\)Page 80 Exercise 13 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|24| = 24
Result
24
Page 80 Exercise 14 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|35.4| = 35.4
Result
35.4
Page 80 Exercise 15 Answer
Account A: −$5.42 … ∣−$5.42∣ = $5.42
Account B: −$35.76 … ∣−$35.76∣ = $35.76
Account B has greater absolute value then Account A, thus Account B has the greater overdrawn amount.
Result
Account B.
Page 80 Exercise 16 Answer
Account A: −$6.47 … ∣−$6.47∣ = $6.47
Account B: −$2.56 … ∣−$2.56∣ = $2.56
Account A has greater absolute value then Account B, thus Account A has the greater overdrawn amount.
Result
Account A.
Page 80 Exercise 17 Answer
Account A: −$32.56 … ∣−$32.56∣ = $32.56
Account B: −$29.12 … ∣−$29.12∣ = $29.12
Account A has greater absolute value then Account B, thus Account A has the greater overdrawn amount.
Result
Account A.
Page 81 Exercise 18 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
∣−46∣ = 46
Result
46
Page 81 Exercise 19 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|0.7| = 0.7
Result
0.7
Page 81 Exercise 20 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|-\(\frac{2}{3}\)| = \(\frac{2}{3}\)
Result
\(\frac{2}{3}\)Page 81 Exercise 21 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
∣−7.35∣ = 7.35
Result
7.35
Page 81 Exercise 22 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|-\(4 \frac{3}{4}\)| = \(4 \frac{3}{4}\)
Result
\(4 \frac{3}{4}\)Page 81 Exercise 23 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
∣−54.5∣ = 54.5
Result
54.5
Page 81 Exercise 24 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|\(27 \frac{1}{4}\)| = \(27 \frac{1}{4}\)
Result
\(27 \frac{1}{4}\)Page 81 Exercise 25 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
∣−13.35∣ = 13.35
Result
13.35
Page 81 Exercise 26 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|14| = 14
Result
14
Page 81 Exercise 27 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
∣−11.5∣ = 11.5
Result
11.5
Page 81 Exercise 28 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
∣−6.3∣ = 6.3
Result
6.3
Page 81 Exercise 29 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|3.75| = 3.75
Result
3.75
Page 81 Exercise 30 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
∣−8.5∣ = 8.5
Result
8.5
Page 81 Exercise 31 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|15| = 15
Result
15
Page 81 Exercise 32 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|-\(6 \frac{3}{4}\)| = \(6 \frac{3}{4}\)
Result
\(6 \frac{3}{4}\)Page 81 Exercise 33 Answer
The absolute value of a number is its distance from 0 on a number line. Since distance is always positive, so is the absolute value.
|-5.3| = 5.3
Result
5.3
Page 81 Exercise 34 Answer
We need to order the number |-12|, |\(11 \frac{3}{4}\)|, |-20.5|, and |2| from least to greatest.
Evaluating the absolute values gives:
|-12| = 12
|\(11 \frac{3}{4}\)| = \(11 \frac{3}{4}\)
|-20.5| = 20.5
|2| = 2
Since 2 < \(11 \frac{3}{4}\) < 12 < 20.5, then the order of the number from least to greatest is:
\(|2|,\left|11 \frac{3}{4}\right|,|-12|,|-20.5|\)Result
\(|2|,\left|11 \frac{3}{4}\right|,|-12|,|-20.5|\)Page 81 Exercise 35 Answer
We need to order the number ∣10∣, ∣−3∣, ∣0∣, and ∣−5.25∣ from least to greatest.
Evaluating the absolute values gives:
|10| = 10
∣−3∣ = 3
|0| = 0
∣−5.25∣ = 5.25
Since 0 < 3 < 5.25 < 10, then the order of the number from least to greatest is:
∣0∣,∣−3∣,∣−5.25∣,∣10∣
Result
∣0∣,∣−3∣,∣−5.25∣,∣10∣
Page 81 Exercise 36 Answer
We need to order the number ∣−6∣, ∣−4∣, ∣11∣, and ∣0∣ from least to greatest.
Evaluating the absolute values gives:
∣−6∣ = 6
∣−4∣ = 4
∣11∣ = 11
∣0∣ = 0
Since 0 < 4 < 6 < 11, then the order of the number from least to greatest is:
∣0∣,∣−4∣,∣−6∣,∣11∣
Result
∣0∣,∣−4∣,∣−6∣,∣11∣
Page 81 Exercise 37 Answer
We need to order the number ∣4∣, ∣−3∣, ∣−18∣, and ∣−3.18∣ from least to greatest.
Evaluating the absolute values gives:
∣4∣ = 4
∣−3∣ = 3
∣−18∣ = 18
∣−3.18∣ = 3.18
Since 3 < 3.18 < 4 < 18, then the order of the number from least to greatest is:
∣−3∣,∣−3.18∣,∣4∣,∣−18∣
Result
∣−3∣,∣−3.18∣,∣4∣,∣−18∣
Page 81 Exercise 38 Answer
As seen in the picture, the stake is marked as zero. Alberto’s horseshoe is 3 feet to the left of the stake, thus the integer which best describes its location is −3. Rebecca’s horseshoe is 2 feet to the right of the stake, thus the integer which best describes its location is 2.
Result
Alberto’s: −3 Rebecca’s: 2
Page 81 Exercise 39 Answer
Even though −3 is less than 2, the person who wins is the person whose horseshoe has the smallest distance from the stake, which is always positive, so we must compare the absolute values of integers −3 and 2.
∣−3∣ = 3
∣2∣ = 2
The absolute value of 2 is less than the absolute value of −3, thus Rebecca’s horseshoe is closer to the stake. Thus, Rebecca wins a point.
Result
Alberto is incorrect. Rebecca’s horseshoe is closer to the stake. Thus, Rebecca wins a point.
Page 81 Exercise 40 Answer
To find the distance from Alberto’s horseshoe to Rebecca’s we must add the distance of Alberto’s horseshoe from the stake and the distance of Rebecca’s horseshoe from the stake because Alberto’s is -3 feet in front of stake and Rebecca’s 2 feet past the stake.
∣−3∣ + ∣2∣ = 3 + 2 = 5
Result
The distance from Alberto’s horseshoe to Rebecca’s is 5 feet.
Page 82 Exercise 41 Answer
The absolute value of a number is its distance from 0 on a number line, since distance is always a positive number so is the absolute value.
The absolute value of a, if a is a positive number is ∣a| = a, and if a is a negative number, the absolute value is ∣a∣ = −a.
The absolute value of a is then different if a is a positive number or a negative number.
Result
If a is a positive number, then ∣a∣ = a and if a is a negative number, then ∣a∣ = −a. The absolute value of a is then different if a is a positive number or a negative number.
Page 82 Exercise 42 Answer
To find out who won the first round of the game we must find the absolute value of -19 and 21, and compare them.
∣−19∣ = 19
∣21∣ = 21
Since 21 has greater absolute value than -19, Leticia won the first round.
Result
Leticia won the first round.
Page 82 Exercise 43 Answer
To find which girl is located farther from the cave entrance we must find the aboslute value of -30 and of -12 since we are comparing distances.
∣−30∣ = 30
∣−12∣ = 12
The absolute value of -30 is greater than the absolute value of -12, thus Ana is farther away from the cave entrance than Chuyen.
Result
Ana is farther away from the cave entrance than Chuyen.
Page 82 Exercise 44 Answer
it is given that Marie’s account balance is −$45.62 and Tom’s account balance is −$42.55.
To determine which balance represents the greater number, we need to determine which number is farther to the right on a number line.
Since −45.62 lies to the left of −45 and −42.44 lies to the right of −45, then −42.55 lies to the right of −45.62. Therefore, −$42.55 is greater than −$45.62 so
Marie’s balance represents the greater number.
To determine which balance represents the lesser amount owed, we need to compare the absolute values:
∣−$45.62∣ = $45.62
∣−$42.55∣ = $42.55
Since $42.44 < $45.62, then Marie’s balance represents the lesser amount owed.
Result
Marie’s balance of −$42.55 is the greater number and the lesser amount owed.
Page 82 Exercise 45 Answer
The Federal Reserve gold vault is located at a depth of ∣−80∣ feet below ground and the treasure at Oak Island is at a depth of ∣−134∣ feet. We must find the absolute values.
∣−80∣ = 80
∣−134∣ = 134
Since ∣−134∣ is greater than ∣−80∣, the Oak Island treasure is farther below ground.
Result
The Oak Island treasure is farther below ground.
Page 82 Exercise 46 Answer
The sea level is marked as zero, so the diver whose location is represented by a number with a lesser absolute value is closest to sea level.
∣−30∣ = 30
∣−42∣ = 42
Since the absolute value of −30 is less than the absolute value of −42, the diver whose location is represented by −30 is closer to sea level.
Result
−30 feet is closer to sea level.
Page 82 Exercise 47a Answer
From the table, the scores are −6, 5, 2, and −3. We need to arrange the scores from least to greatest.
Negative numbers are always smaller than positive numbers so −6 and −3 are the smaller numbers and 5 and 2 are the larger numbers.
Since −6 lies farther left on a number line than −3, then −6 < −3:
−6,−3, _, _
Since 2 lies farther left on a number line than 5, then 2 < 5. The order of the scores is then:
−6,−3,2,5
Result
−6,−3,2,5
Page 82 Exercise 47b Answer
From PART A, we know the scores from least to greatest is:
−6,−3,2,5
If the least score is the winning score, then the score of −6 is the winning score. Therefore, Kate won the first round.
Result
Kate