Question

absolute value: number lines

circle the correct description of the number line.
a. number line: -2, -1, 0, 1, 2 with two dots |1| and its opposite; |1| and |1|; 1 and |-1|
b. number line: -1, 0, 1 with two dots |0.75| and its opposite; |-0.75| and |0.75|; 0.75 and |-0.75|
c. number line: -2, -1, 0, 1, 2 with a dot at 2 |-2| and its opposite; |-2| and |2|; -2 and |-2|
d. number line: -5, -4, ..., 5 with a dot at -4 and 4 |-4| and its opposite; |-4| and |4|; 4 and |-4|

circle the number farthest from 0.
e. $|-1|$, $\left|\\!-\frac{4}{3}\
ight|$, $\left|\frac{2}{3}\
ight|$
f. $22$, $|-3|$, $|17|$
g. $|-3.5|$, $|2.5|$, $2$
h. $|-10|$, $5$, $21$
i. $\left|\\!-1\frac{1}{5}\
ight|$, $|1|$, $|-1.4|$
j. $-10$, $-11$, $|-5|$

circle the number farther to the right of the bold number on a number line.
k. $-9$, $|-8|$, $\left|\frac{7}{1}\
ight|$ bold: $7$?
l. $-\frac{1}{2}$, $\left|\frac{1}{2}\
ight|$, $-1$, $-0.75$ bold: $-\frac{1}{2}$?
m. $-29$, $-26$, $|1|$, $-27$ bold: $-26$?
n. $|-21|$, $|3|$, $-12$ bold: $11$?
o. $|-25|$, $-45$, $35$ bold: $|30|$?
p. $0$, $|24|$, $|-51|$ bold: $|-42|$?

circle the number farther to the left of the bold number on a number line.
q. $-2$, $|-15|$, $8$ bold: $|-8|$?
r. $\frac{2}{3}$, $\frac{1}{6}$, $-\frac{2}{3}$ bold: $-\frac{1}{3}$?
s. $|-17|$, $0$, $18$ bold: $|-16|$?
t. $|-0.8|$, $\left|\frac{1}{4}\
ight|$, $1$ bold: $0.75$?
u. $|-7|$, $5$, $|-3|$ bold: $|5|$?
v. $-11$, $12$, $|-10|$ bold: $-9$?

Answer:

Part a: Correct Description of Number Line

To determine the correct description, recall that the absolute value of a number \(|x|\) is its distance from 0, and the opposite of \(|x|\) (if \(x\) is positive) is \(-|x|\). The number line in part a has points at \(-1\) and \(1\).

• \(|1|\) is \(1\), and its opposite is \(-1\).
• \(|1|\) and \(|1|\) would be the same point, so incorrect.
• \(1\) and \(|-1|\) (which is \(1\)) are the same, so incorrect.

Thus, the correct description is \(|1|\) and its opposite.

Part b: Correct Description of Number Line

The number line in part b has points at \(-0.75\) (opposite of \(0.75\)) and \(0.75\).

• \(|0.75| = 0.75\), and its opposite is \(-0.75\).
• \(|-0.75| = 0.75\) and \(|0.75| = 0.75\) are the same, so incorrect.
• \(0.75\) and \(|-0.75| = 0.75\) are the same, so incorrect.

Thus, the correct description is \(|0.75|\) and its opposite.

Part c: Correct Description of Number Line

The number line in part c has a point at \(2\). \(|-2| = 2\), and its opposite is \(-2\), but the line shows \(2\).

• \(|-2| = 2\) and \(|2| = 2\) (same value, correct as they represent the same distance from 0).
• \(|-2|\) and its opposite would be \(2\) and \(-2\), but the line only has \(2\), so incorrect.
• \(-2\) and \(|-2| = 2\) are different, but the line shows \(2\), so incorrect.

Thus, the correct description is \(|-2|\) and \(|2|\).

Part d: Correct Description of Number Line

The number line in part d has a point at \(4\). \(|-4| = 4\), and its opposite is \(-4\), but the line shows \(4\).

• \(|-4| = 4\) and \(|4| = 4\) (same value, correct as they represent the same distance from 0).
• \(|-4|\) and its opposite would be \(4\) and \(-4\), but the line only has \(4\), so incorrect.
• \(4\) and \(|-4| = 4\) are the same, so incorrect.

Thus, the correct description is \(|-4|\) and \(|4|\).

Part e: Number Farthest from 0

Calculate absolute values:

• \(|-1| = 1\)
• \(|-\frac{4}{3}| = \frac{4}{3} \approx 1.33\)
• \(|\frac{2}{3}| \approx 0.67\)

Compare: \(\frac{4}{3} > 1 > \frac{2}{3}\). Thus, \(|-\frac{4}{3}|\) is farthest from 0.

Part f: Number Farthest from 0

Absolute values:

• \(|22| = 22\)
• \(|-3| = 3\)
• \(|17| = 17\)

Compare: \(22 > 17 > 3\). Thus, \(22\) is farthest from 0.

Part g: Number Farthest from 0

Absolute values:

• \(|-3.5| = 3.5\)
• \(|2.5| = 2.5\)
• \(|2| = 2\)

Compare: \(3.5 > 2.5 > 2\). Thus, \(|-3.5|\) is farthest from 0.

Part h: Number Farthest from 0

Absolute values:

• \(|-10| = 10\)
• \(|5| = 5\)
• \(|21| = 21\)

Compare: \(21 > 10 > 5\). Thus, \(21\) is farthest from 0.

Part i: Number Farthest from 0

Calculate absolute values:

• \(|-1\frac{1}{5}| = 1.2\)
• \(|1| = 1\)
• \(|-1.4| = 1.4\)

Compare: \(1.4 > 1.2 > 1\). Thus, \(|-1.4|\) is farthest from 0.

Part j: Number Farthest from 0

Absolute values:

• \(|-10| = 10\)
• \(|-11| = 11\)
• \(|-5| = 5\)

Compare: \(11 > 10 > 5\). Thus, \(-11\) (since \(|-11| = 11\)) is farthest from 0.

Part k: Number Farther to the Right of Bold \(7\)

Numbers: \(-9\), \(|-8| = 8\), \(|\frac{7}{1}| = 7\).
On a number line, \(8 > 7 > -9\). Thus, \(|-8|\) is farther right.

Part l: Number Farther to the Right of Bold \(-\frac{1}{2}\)

Numbers: \(|\frac{1}{2}| = 0.5\), \(-1\), \(-0.75\).
On a number line: \(0.5 > -0.5 > -0.75 > -1\). Thus, \(|\frac{1}{2}|\) is farther right.

Part m: Number Farther to the Right of Bold \(-26\)

Numbers: \(-29\), \(|1| = 1\), \(-27\).
On a number line: \(1 > -26 > -27 > -29\). Thus, \(|1|\) is farther right.

Part n: Number Farther to the Right of Bold \(11\)

Numbers: \(|-21| = 21\), \(|3| = 3\), \(-12\).
On a number line: \(21 > 11 > 3 > -12\). Thus, \(|-21|\) is farther right.

Part o: Number Farther to the Right of Bold \(|30| = 30\)

Numbers: \(|-25| = 25\), \(-45\), \(35\).
On a number line: \(35 > 30 > 25 > -45\). Thus, \(35\) is farther right.

Part p: Number Farther to the Right of Bold \(|-42| = 42\)

Numbers: \(0\), \(|24| = 24\), \(|-51| = 51\).
On a number line: \(51 > 42 > 24 > 0\). Thus, \(|-51|\) is farther right.

Part q: Number Farther to the Left of Bold \(|-8| = 8\)

Numbers: \(-2\), \(|-15| = 15\), \(8\).
On a number line: \(-2 < 8 < 15\). Thus, \(-2\) is farther left.

Part r: Number Farther to the Left of Bold \(-\frac{1}{3}\)

Numbers: \(\frac{2}{3}\), \(\frac{1}{6}\), \(-\frac{2}{3}\).
On a number line: \(-\frac{2}{3} < -\frac{1}{3} < \frac{1}{6} < \frac{2}{3}\). Thus, \(-\frac{2}{3}\) is farther left.

Part s: Number Farther to the Left of Bold \(|-16| = 16\)

Numbers: \(|-17| = 17\), \(0\), \(18\).
On a number line: \(0 < 16 < 17 < 18\). Thus, \(0\) is farther left.

Part t: Number Farther to the Left of Bold \(0.75\)

Numbers: \(|-0.8| = 0.8\), \(|\frac{1}{4}| = 0.25\), \(1\).
On a number line: \(0.25 < 0.75 < 0.8 < 1\). Thus, \(|\frac{1}{4}|\) is farther left.

Part u: Number Farther to the Left of Bold \(|5| = 5\)

Numbers: \(|-7| = 7\), \(5\), \(|-3| = 3\).
On a number line: \(3 < 5 < 7\). Thus, \(|-3|\) is farther left.

Part v: Number Farther to the Left of Bold \(-9\)

Numbers: \(-11\), \(12\), \(|-10| = 10\).
On a number line: \(-11 < -9 < 10 < 12\). Thus, \(-11\) is farther left.

Final Answers (Key Selections):

a. \(|1|\) and its opposite
b. \(|0.75|\) and its opposite
c. \(|-2|\) and \(|2|\)
d. \(|-4|\) and \(|4|\)
e. \(|-\frac{4}{3}|\)
f. \(22\)
g. \(|-3.5|\)
h. \(21\)
i. \(|-1.4|\)
j. \(-11\)
k. \(|-8|\)
l. \(|\frac{1}{2}|\)
m. \(|1|\)
n. \(|-21|\)
o. \(35\)
p. \(|-51|\)
q. \(-2\)
r. \(-\frac{2}{3}\)
s. \(0\)
t. \(|\frac{1}{4}|\)
u. \(|-3|\)
v. \(-11\)