Chapter 2 Integers And Rational Numbers
Vocabulary Review
Page 107 Exercise 1 Answer
A point on a coordinate plane is represented by an ordered pair.
Coordinate plane is made up of two perprendicular number lines who meet at the origin, the x-axis and the y-axis. Each point on a coordinate plane has a unique ordered pair which describes its location.
For example, let’s take point A whose x-coordinate is 3 and y-coordinate is −4.
A(3,−4)
Result
Ordered pair.
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Page 107 Exercise 2 Answer
The opposite of a positive integer is a negative integer.
For example, the opposite number of 3 is −3, and the opposite number of −18 is 18.
Result
Opposite.
Page 107 Exercise 3 Answer
A rational number is any number that can be written as the quotient of two integers.
For example, a fraction \(\frac{3}{4}\) is a rational number because it can be written as a quotient 3 ÷ 4, or a mixed number \(7 \frac{1}{4}\), which can be written as \(\frac{29}{4}\) and further as 29 ÷ 4, or a decimal such as 0.25, which can be written as a fraction \(\frac{1}{4}\) and further as 1 ÷ 4.
Result
Rational number.
Page 108 Exercise 1 Answer
We need to plot \(\frac{3}{4}\) on the given number line.
The number line is divided into tenths so it would be easier to graph if we converted \(\frac{3}{4}\) to a decimal:
\(\frac{3}{4}=\frac{75}{100}=0.75\)The location of \(\frac{3}{4}\) is then halfway between 0.7 and 0.8, which are the 7th and 8th tick marks to the right of 0 on the number line:
Result
Plot a point halfway between the 7th and 8th tick marks to the right of 0 on the number line.
Page 108 Exercise 2 Answer
We need to plot –\(\frac{2}{5}\) on the given number line.
The number line is divided into tenths so it would be easier to graph if we converted –\(\frac{2}{5}\) to a decimal:
\(-\frac{2}{5}=-\frac{4}{10}=-0.4[\latex]
The location of –[latex]\frac{2}{5}\) is then −0.4, which is the 4th tick mark to the left of 0 on the number line:
Result
Plot a point at −0.4, which is the 4th tick mark to the left of 0 on the number line.
To plot –\(\frac{2}{5}\) divide the first unit to the left of zero into five equal parts and mark the second from the left or find −0,4 on the number line since –\(\frac{2}{5}\) = −0,4.
Result
Open to see the solution.
Page 108 Exercise 3 Answer
To plot 0.5 divide the first unit to the right of zero into half, since 0.5 = \(\frac{1}{2}\).
Result
Open to see the solution.
The given number line is divided into tenths so to plot 0.5 on the number line, plot a point halfway between 0 and 1, which is the 5th tick mark to the right of 0:
Result
Plot a point halfway between 0 and 1.
Page 108 Exercise 4 Answer
We need to compare 0.25 and \(\frac{1}{4}\)
Two rational numbers are easier to compare if they are both fractions or both decimals.
Converting 0.25 to a fraction gives:
\(0.25=\frac{25}{100}=\frac{1}{4}\)
Therefore, 0.25 = \(\frac{1}{4}\).
Result
0.25 = \(\frac{1}{4}\)
Fraction \(\frac{1}{4}\) can be written as a quotient 1 ÷ 4 which than gives 0.25. Thus
0.25 = \(\frac{1}{4}\).
Result
0.25 = \(\frac{1}{4}\)
Page 108 Exercise 5 Answer
We need to compare \(1 \frac{5}{8}\) and 1.6.
Two rational numbers are easier to compare if they are both fractions or both decimals.
Dividing 5 and 8 gives:
We can then rewrite \(1 \frac{5}{8}\) as the decimal 1.625.
Since 1.625 > 1.6, then \(1 \frac{5}{8}\) > 1.6
Result
\(1 \frac{5}{8}\) < 1.6
Fraction \(1 \frac{5}{8}\) is equal to \(\frac{13}{8}\), which can be further written as a quotient 13 ÷ 8 which is equal to 1.625. Thus \(1 \frac{5}{8}\) > 1.6.
Result
\(1 \frac{5}{8}\) > 1.6.
Page 108 Exercise 6 Answer
We need to compare 3.65 and \(3 \frac{3}{4}\).
Two rational numbers are easier to compare if they are both fractions or both decimals.
Converting \(3 \frac{3}{4}\) to a decimal gives:
\(3 \frac{3}{4}=3 \frac{75}{100}=3.75\)Since 3.65 < 3.75, then 3.65 < \(3 \frac{3}{4}\)
Result
3.65 < \(3 \frac{3}{4}\)
Fraction \(3 \frac{3}{4}\) is equal to \(\frac{15}{4}\) which can be written as a quotient 15 ÷ 4 which is equal to 3.75. Thus,
3.65 < \(3 \frac{3}{4}\).
Result
3.65 < \(3 \frac{3}{4}\)
Page 108 Exercise 7 Answer
To compare two fractions we must rewrite them with the same denominator. To do that first we need to find a common multiple of 3 and 4.
\(-\frac{2}{3}=-\frac{8}{12} \quad-\frac{3}{4}=-\frac{9}{12}\)Thus,
\(-\frac{8}{12}>-\frac{9}{12} \text { so }-\frac{2}{3}>-\frac{3}{4}\)Result
\(-\frac{2}{3}>-\frac{3}{4}\)Page 108 Exercise 1 Answer
The integer for point A is 3 since A is three units to the right from zero. The opposite number is −3.
Result
The integer is 3 and its opposite is −3.
Page 108 Exercise 2 Answer
The integer for point B is −1 since point B is one unit left from zero. The opposite number is 1.
Result
The integer is −1 and its opposite is 1.
Page 108 Exercise 3 Answer
The integer for point C is 6 since point C is six units right from zero. Its opposite number is −6.
Result
The integer is 6 and its opposite is −6.
Page 108 Exercise 4 Answer
The integer for point D is −7 since it is seven units left from zero. Its opposite number is 7.
Result
The integer is −7 and its opposite number is 7.
Page 108 Exercise 5 Answer
The integer for point E is −5 since it is five units left from zero. Its opposite number is 5.
Result
The integer is −5 and its opposite number is 5.
Page 108 Exercise 6 Answer
The integer for point F is 1 since it is one unit right from zero. Its opposite number is −1.
Result
The integer is 1 and its opposite is −1.
Page 109 Exercise 1 Answer
Absolute value of a number is either positive or equal to zero. It is equal to zero if and only if the number is zero. The absolute value of a number represents the distance of that number from zero on the number line.
−9 is nine units left from zero on the number line, thus
∣−9∣ = 9.
Result
The absolute value of −9 is 9.
Page 109 Exercise 2 Answer
Absolute value of a number is either positive or equal to zero. It is equal to zero if and only if the number is zero. The absolute value of a number represents the distance of that number from zero on the number line.
−2 is two units left from zero on the number line, thus
∣−2∣ = 2.
Result
The absolute value of −2 is 2.
Page 109 Exercise 3 Answer
Absolute value of a number is either positive or equal to zero. It is equal to zero if and only if the number is zero. The absolute value of a number represents the distance of that number from zero on the number line.
4 is four units right from zero on the number line, thus
∣4∣ = 4.
Result
The absolute value of 4 is 4.
Page 109 Exercise 4 Answer
Absolute value of a number is either positive or equal to zero. It is equal to zero if and only if the number is zero. The absolute value of a number represents the distance of that number from zero on the number line.
Result
The solution is −10.
Page 109 Exercise 5 Answer
First, find the absolute values.
∣−3∣ = 3
∣−2∣ = 2
∣10∣ = 10
Ordered from least to greatest:
2, 3, 10 → ∣−2∣, ∣−3∣, ∣10∣
Result
∣−2∣, ∣−3∣, ∣10∣
Page 109 Exercise 6 Answer
First, find the absolute values.
∣−7∣ = 7
∣0∣ = 0
∣−5∣ = 5
Ordered from least to greatest:
0, 5, 7 → ∣0∣, ∣−5∣, ∣−7∣
Result
∣0∣, ∣−5∣, ∣−7∣
Page 109 Exercise 7 Answer
First, find the absolute values.
∣−18.5∣ = 18.5
∣18∣ = 18
∣−12.5∣ = 12.5
Ordered from least to greatest:
12.5, 18, 18.5 → ∣−12.5∣, ∣18∣, ∣−18.5∣
Result
∣−12.5∣, ∣18∣, ∣−18.5∣
Page 109 Exercise 8 Answer
First, find the absolute values.
∣26∣ = 26
∣−20∣ = 20
∣−24.5∣ = 24.5
Ordered from least to greatest:
20, 24.5, 26 → ∣−20∣, ∣−24.5∣, ∣26∣
Result
∣−20∣, ∣−24.5∣, ∣26∣
Page 109 Exercise 1 Answer
To give the ordered pair for point U follow the grid lines directly from the point to the y-axis to find the y-coordinate, which is 2.5. Since the point is in the y-axis, its x-coordinate is zero.
Result
U(0,2.5)
Page 109 Exercise 2 Answer
To give the ordered pair for point V follow the grid lines directly from the point to the x -axis to find the x-coordinate, which is −2. Follow the grid lines directly from the point to the y – axis to find the y-coordinate, 1.5.
Result
V(−2,1.5)
Page 109 Exercise 3 Answer
To give the ordered pair for point W follow the grid lines directly from the point to the x -axis to find the x-coordinate, which is −4. Follow the grid lines directly from the point to the y -axis to find the y-coordinate, −1.
Result
W(−4,−1)
Page 109 Exercise 4 Answer
To give the ordered pair for point X follow the grid lines directly from the point to the x -axis to find the x-coordinate, which is 2.5. Since the point is in the x-axis, its y-coordinate is zero.
Result
X(2.5,0)
Page 109 Exercise 5 Answer
To give the ordered pair for point Y follow the grid lines directly from the point to the x-axis to find the x-coordinate, which is 2. Follow the grid lines directly from the point to the y-axis to find the y-coordinate, −1.5.
Result
Y(2,−1.5)
Page 109 Exercise 6 Answer
To give the ordered pair for point Z follow the grid lines directly from the point $\textbf{to the x-axis to find the x-coordinate, which is −1.5. Follow the grid lines directly from the point to the y-axis to find the y-coordinate, −3.
Result
Z(−1.5,−3)
Page 110 Exercise 1 Answer
To find the length of side BC first find the ordered pairs which represent them. The ordered pair for point B is B(−1,2), and the ordered pair for point C is C(−1,1). The points are in the same quadrant, Quadrant II and have the same x-coordinates. To find the distance from B to C, subtract the absolute values of their y-coordinates.
∣2∣ − ∣1∣ = 2 − 1 = 1 unit
Result
The length of side BC is 1 unit.
Page 110 Exercise 2 Answer
To find the length of side CD first find the ordered pairs which represent them. The ordered pair for point D is C(−1,1), and the ordered pair for point D is D(1,1). The points are not in the same quadrant and have the same y-coordinates. To find the distance from C to D, add the absolute values of their x-coordinates.
∣−1∣ + ∣1∣ = 1 + 1 = 2 units
Result
The length of side CD is 2 units.
Page 110 Exercise 3 Answer
To find the length of side DE first find the ordered pairs which represent them. The ordered pair for point D is D(1,1), and the ordered pair for point E is E(1,−2). The points are not in the same quadrant and have the same x-coordinates. To find the distance from D to E, add the absolute values of their y-coordinates.
∣1∣ + ∣−2∣ = 1 + 2 = 3 units
Result
The length of side DE is 3 units.
Page 110 Exercise 4 Answer
To find the length of side EF first find the ordered pairs which represent them. The ordered pair for point E is E(1,−2), and the ordered pair for point F is F(−3,−2). The points are not in the same quadrant and have the same y-coordinates. To find the distance from E to F, add the absolute values of their x-coordinates.
∣−3∣ + ∣1∣ = 3 + 1 = 4 units
Result
The length of side EF is 4 units.
Page 110 Exercise 5 Answer
To find the length of side FA first find the ordered pairs which represent them. The ordered pair for point F is F(−3,−2), and the ordered pair for point A is A(−3,2). The points are not in the same quadrant and have the same x-coordinates. To find the distance from F to A, add the absolute values of their y-coordinates.
∣−2∣ + ∣2∣ = 2 + 2 = 4 units
Result
The length of side FA is 4 units.
Page 110 Exercise 6 Answer
The perimeter of polygon ABCDEF is the sum of the lengths of all of its sides.
AB + BC + CD + DE + EF + FA = 2 + 1 + 2 + 3 + 4 + 4 = 16 units
Result
The perimeter of ABCDEF is 16 units.
Page 110 Exercise 7 Answer
To draw a polygon QRST first plot the points Q(−4,−1), R(−4,5), S(2,5), and (2,−1). To do that, find the x-coordinates on the x-axis and find the y-coordinates on the y-axis. Follow the grid lines from each axis for each of the points to where they meet and mark that spot, these are the vertices of the polygon. Connect the adjacent vertices to draw a polygon QRST.
Result
Plot the points Q(−4,−1), R(−4,5), S(2,5), and (2,−1) and connect the adjacent vertices to draw a polygon QRST.
Page 110 Exercise 8 Answer
Polygon QRST is a square if all of its sides are of equal length.
QR = ∣5∣ + ∣−1∣ = 5 + 1 = 6
RS = ∣−4∣ + ∣2∣ = 4 + 2 = 6
ST = ∣5∣ + ∣−1∣ = 5 + 1 = 6
TQ = ∣−4∣ + ∣2∣ = 4 + 2 = 6
All sides of the polygon are of equal length, so the polygon is a square.
Result
Polygon QRST has sides that are of equal length so the polygon is a square.