Download Integers and Absolute Value - Blue Problems

Integers and Absolute Value - Blue Problems
1. Decide whether the statement is true or false. Explain your reasoning.
-|-a| is always negative.
Order the numbers from least to greatest.
2. 1, -2, -(-3), |-4|, -|5|, -|-6|, -(-|-7|)
3. -|22|, -|-17|, 19, -|-(-21)|, 26, -18, -(-17)
Evaluate the expression when a = -6 and b = 9.
4. |-a| + |-b|
5. |-b| - (-a)
6. |b + (-a)|
7. -|b – (-a)|
Tell whether x is positive integer or a negative integer.
8. x = -|-x |
9. - x = -|-x|
10. - x = |-x|
11. -x = -|x |
Adding Integers - Blue Problems
REAL-WORLD PROBLEM SOLVING
Caving
Caving is a sport for people who enjoy maneuvering tight spaces in the dark. It is not
an activity for people who are unfit or claustrophobic. Negotiating the inside of a cave
may involve walking through large passages, climbing up and down steep inclines,
crawling through chambers with low ceilings, and slithering through tunnels so narrow
that the human body can barely fit. For safety reasons, caving should never be done
alone, without proper equipment and clothing, or without an experienced caver
present.
In Exercise 1-3, use the following information.
You and three friends are about to venture deep inside a cave to visit the “rain room”
you have heard about. To keep track of how deep you are, you decide to estimate the
vertical change you go through at each step along the way. You step through the cave
entrance and follow a long, narrow passage that you estimate takes you down about
85 feet. At this point, there is what appears to be 4-foot drop and the passage turns to
the left. You estimate that the passage gradually descends about another 55 feet, then
makes a steep drop of about 27 feet into an open chamber. The floor of the chamber
seems to rise about 9 feet to the opposite wall where you enter a narrow tunnel with
several twists and turns. You estimate that the tunnel takes you down about 22 feet
where you enter the rain room. This is a small room about 10 feet in diameter and has
about 3 inches of water on the floor. The rain room is named because water drips off
of its 5-foot high ceiling to an extent that resembles a heavy rain storm! After you check
out the rain room for several minutes, you begin to head back toward the cave
entrance.
1. Write an expression that represents your vertical change in altitude from the cave
entrance to the rain room as a sum of integers.
2. Evaluate the sum you wrote in Exercise 1.
3. As you emerge from the cave, you meet some cavers who are preparing to enter. They
tell you that the rain room is actually 203 feet lower than the entrance to the cave. Add
this number to your answer from Exercise 2 to see how close the total of the vertical
changes you estimated is to the actual change.
4. Try different integer values of a and b in the expressions |a + b| and |a| + |b|.
Determine what types of integers make the equation |a + b| and |a| + |b| true.
Find the sum.
5. -347 + 67 + (-1243) + 4343 + (-2351)
6. -5760 + 456 + (-992) + (-41) + (-656) + 2208
In the statement, a and b are nonzero integers. Explain what must be true about the value
of a and b.
7. |a| + b = a + b
8. a + |b| = |a| + b
9. |a + b|  |a| + |b|
10. |a + b| = |a| + |b|
11. a + b = 0
12. a + b = 1
13. The sum a + b is a negative integer.
14. The sum a + (-b) is a positive integer.
15. Each Monday Mr. Berntson’s class plays the Integer Game. Students play in pairs using
a set of cards. Each card has a negative or a positive number written on it. The object
of the game is to pick six cards so that the numbers written on those cards have a sum
of 0.
These are the cards that are left that Alisa can choose from:
A: What are the numbers on the cards that she should pick?
B: What if you changed the rules of the game so that any number of cards could be
used. What's the most number of cards you can pick out so that the sum is 0?
Integers and Absolute Value - Blue Solutions
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True. |-a| is always positive, thus -|-a|must be negative.
-|-6|, -|5|, -2, 1, -(-3), |-4|, , -(-|-7|)
-|22|, -|-(-21)|, -18,-|-17|, -(-17),19, 26
15
3
15
-3
Negative
Positive
Negative
Positive
Adding Integers - Blue Solutions
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-85 + (-4) + (-55) + (-27) + 9 + (-22)
-184
203 + (-184) = 19; The total of the estimates is 19 feet less than the actual vertical distance.
|a + b| = |a| + |b| when a = 0, when b = 0, when a and b are both positive, and when a
and b are both negative.
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a is positive, b can be any integer.
(1) Both a and b are positive, or
(2) a and b are negative and a = b.
One of the integers a and b is positive, the other is negative.
(1) Both a and b are positive, or
(2) Both a and b are negative.
a and b are opposites.
One of the integers a and b is positive, the other is negative. The absolute value of the
positive integer is 1 greater than the absolute value of the negative integer.
(1) Both a and b are negative, or
(2) One is negative, the other is positive, and the absolute value of the negative integer is
greater than the absolute value of the positive integer.
(1) a is positive and b is negative, or
(2) a and b are both positive and a is greater than b, or
(3) a and b are both negative and the absolute value of b is greater than the absolute value
of a.
A:
1) 10, -2, -11, -8, 5, 6
2) -2, -3,-6, 2, 4, 5
3) 23, -8, -3, -6, -11, 5
4) 20, 2, -11, -6, -3, -2
5) 4, 23, -2, -6, -8, -11
6) 20, 4, -3, -2, -11, -8
7) 10, 6, 4,-11, -6, -3
8) 10, 4, 2, -6, -2, -8
B:
The most is 9 numbers
6, -6, 2, -2, 4, 5, 10, -8, -11