Question

1. higher order thinking let a = any rational number. is the absolute value of a different if a is a positive number or a negative number? explain.
2. communicate and justify tyrell and leticia are playing a game. after the first round of the game, tyrells score was - 19, and leticias score was 21. the score with the greater absolute value wins each round. who won the first round? explain.
3. use patterns and structure ana and chuyen are exploring underwater sea life while on a helmet diving adventure. anas location relative to sea level is - 30 feet, and chuyens location is - 12 feet. which girl is located farther from sea level, and how much farther?
4. maries account balance is -$45.62. toms account balance is -$42.55. which balance is greater? which balance represents the lesser amount owed?
5. in new york, the federal reserve gold vault is located at a depth of | - 80| feet. the treasure at oak island is believed to be at a depth of | - 134| feet. which is farther below ground, the gold vault or the oak island treasure? how far below ground it is?
6. two scuba divers are swimming below sea level. the locations of the divers can be represented by - 30 feet and - 42 feet. which measure represents the location that is closer to sea level?
7. the table at the right shows the scores at the end of the first round of a golf tournament. the scores are relative to par. 6.ns0.1.4, 6.ns0.1.3

part a
par is represented as 0. using absolute value, show the distance each score is from par.
part b
the golfer with the least score wins the round. who won the first round of the tournament? explain.
golfer kate sam lisa carlos
score - 6 5 2 - 3

Explanation:

Step1: Recall absolute - value definition

The absolute value of a number \(x\), denoted as \(|x|\), is the non - negative value of \(x\) without regard to its sign. Mathematically, \(|x|=

$$\begin{cases}x, & x\geq0\\ - x, & x < 0\end{cases}$$

\).

Step2: Analyze positive and negative cases for question 41

If \(a>0\), then \(|a| = a\). If \(a < 0\), then \(|a|=-a\), which is a positive number. For example, if \(a = 5\), \(|5| = 5\); if \(a=-5\), \(| - 5|=5\). So the absolute value of a rational number \(a\) is the same whether \(a\) is positive or negative.

Step3: Solve question 42

Tyrell's score is \(s_T=-19\), so \(|s_T| = |-19| = 19\). Leticia's score is \(s_L = 21\), so \(|s_L|=|21| = 21\). Since \(21>19\), Leticia won the first round because her score has a greater absolute value.

Step4: Solve question 43

Ana's location is \(d_A=-30\) feet and Chuyen's location is \(d_C=-12\) feet. The absolute value of Ana's distance from sea - level is \(|d_A| = |-30| = 30\) feet, and the absolute value of Chuyen's distance from sea - level is \(|d_C|=|-12| = 12\) feet. Since \(30>12\), Ana is located farther from sea level. The difference in distances is \(30 - 12=18\) feet.

Step5: Solve question 44

Marie's account balance is \(b_M=-45.62\), so \(|b_M| = |-45.62| = 45.62\). Tom's account balance is \(b_T=-42.55\), so \(|b_T|=|-42.55| = 42.55\). Since \(42.55<45.62\), Tom's balance is greater (because \(-42.55>-45.62\)), and Tom represents the lesser amount owed.

Step6: Solve question 45

The depth of the Federal Reserve gold vault is \(d_{gv}=|-80| = 80\) feet. The depth of the Oak Island treasure is \(d_{ot}=|-134| = 134\) feet. Since \(134>80\), the Oak Island treasure is farther below ground, and it is \(134 - 80 = 54\) feet farther below ground.

Step7: Solve question 46

The first diver's location is \(d_1=-30\) feet, so \(|d_1| = |-30| = 30\) feet. The second diver's location is \(d_2=-42\) feet, so \(|d_2|=|-42| = 42\) feet. Since \(30<42\), the location represented by \(-30\) feet is closer to sea level.

Step8: Solve question 47 Part A

Kate's score \(s_K=-6\), \(|s_K| = |-6| = 6\). Sam's score \(s_S = 5\), \(|s_S|=|5| = 5\). Lisa's score \(s_L = 2\), \(|s_L|=|2| = 2\). Carlos's score \(s_C=-3\), \(|s_C| = |-3| = 3\).

Step9: Solve question 47 Part B

The scores are \(s_K=-6\), \(s_S = 5\), \(s_L = 2\), \(s_C=-3\). Since \(-6<-3<2<5\), Kate won the first round because she has the least score.

Answer:

1. No, the absolute value of a rational number \(a\) is the same whether \(a\) is positive or negative.
2. Leticia won. \(|-19| = 19\), \(|21| = 21\), and \(21>19\).
3. Ana is farther from sea level by 18 feet. \(|-30| = 30\), \(|-12| = 12\), \(30-12 = 18\).
4. Tom's balance is greater. Tom represents the lesser amount owed. \(|-45.62| = 45.62\), \(|-42.55| = 42.55\), \(-42.55>-45.62\).
5. The Oak Island treasure is farther below ground by 54 feet. \(|-80| = 80\), \(|-134| = 134\), \(134 - 80 = 54\).
6. The location represented by \(-30\) feet is closer to sea level. \(|-30| = 30\), \(|-42| = 42\), \(30<42\).

47 Part A: Kate: 6, Sam: 5, Lisa: 2, Carlos: 3.
47 Part B: Kate won. Her score \(-6\) is the least among \(-6,5,2,-3\).