3.1 Introduction to integers  (Page 5/9)

 Page 5 / 9

Translate into an expression with integers:

The football team had a gain of $5\phantom{\rule{0.2em}{0ex}}\text{yards.}$

5 yards

Translate into an expression with integers:

The scuba diver was $30\phantom{\rule{0.2em}{0ex}}\text{feet}$ below the surface of the water.

−30 feet

Key concepts

• Opposite Notation
• $-a$ means the opposite of the number $a$
• The notation $-a$ is read the opposite of $a.$
• Absolute Value Notation
• The absolute value of a number $n$ is written as $|n|$ .
• $|n|\ge 0$ for all numbers.

Practice makes perfect

Locate Positive and Negative Numbers on the Number Line

In the following exercises, locate and label the given points on a number line.

1. $\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-2\phantom{\rule{1em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}-5$

1. $\phantom{\rule{0.2em}{0ex}}5\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-5\phantom{\rule{1em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}-2$
1. $\phantom{\rule{0.2em}{0ex}}-8\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}8\phantom{\rule{1em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}-6$

1. $\phantom{\rule{0.2em}{0ex}}-7\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}7\phantom{\rule{1em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}-1$

Order Positive and Negative Numbers on the Number Line

In the following exercises, order each of the following pairs of numbers, using $<$ or $\text{>.}$

1. $\phantom{\rule{0.2em}{0ex}}9\text{__}4\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-3\text{__}6\phantom{\rule{0.2em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}-8\text{__}-2\phantom{\rule{0.2em}{0ex}}$
4. $\phantom{\rule{0.2em}{0ex}}1\text{__}-10$

1. >
2. <
3. <
4. >
1. $\phantom{\rule{0.2em}{0ex}}6\text{__}2;\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-7\text{__}4;\phantom{\rule{0.2em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}-9\text{__}-1;\phantom{\rule{0.2em}{0ex}}$
4. $\phantom{\rule{0.2em}{0ex}}9\text{__}-3$
1. $\phantom{\rule{0.2em}{0ex}}-5\text{__}1;\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-4\text{__}-9;\phantom{\rule{0.2em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}6\text{__}10;\phantom{\rule{0.2em}{0ex}}$
4. $\phantom{\rule{0.2em}{0ex}}3\text{__}-8$
1. <
2. >
3. <
4. >
1. $\phantom{\rule{0.2em}{0ex}}-7\text{__}3;\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-10\text{__}-5;\phantom{\rule{0.2em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}2\text{__}-6;\phantom{\rule{0.2em}{0ex}}$
4. $\phantom{\rule{0.2em}{0ex}}8\text{__}9$

Find Opposites

In the following exercises, find the opposite of each number.

1. $\phantom{\rule{0.2em}{0ex}}2\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-6$
1. −2
2. 6
1. $\phantom{\rule{0.2em}{0ex}}9\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-4$
1. $\phantom{\rule{0.2em}{0ex}}-8\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}1$
1. 8
2. −1
1. $\phantom{\rule{0.2em}{0ex}}-2\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}6$

In the following exercises, simplify.

$-\left(-4\right)$

4

$-\left(-8\right)$

$-\left(-15\right)$

15

$-\left(-11\right)$

In the following exercises, evaluate.

$-m\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}m=3\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}m=-3$

1. −3
2. 3

$-p\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}p=6\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}p=-6$

$-c\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}c=12\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}c=-12$

1. −12;
2. 12

$-d\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

1. $\phantom{\rule{0.2em}{0ex}}d=21\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}d=-21$

Simplify Expressions with Absolute Value

In the following exercises, simplify each absolute value expression.

1. $\phantom{\rule{0.2em}{0ex}}|7|\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}|-25|\phantom{\rule{1em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}|0|$
1. 7
2. 25
3. 0
1. $\phantom{\rule{0.2em}{0ex}}|5|\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}|20|\phantom{\rule{1em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}|-19|$
1. $\phantom{\rule{0.2em}{0ex}}|-32|\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}|-18|\phantom{\rule{1em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}|16|$
1. 32
2. 18
3. 16
1. $\phantom{\rule{0.2em}{0ex}}|-41|\phantom{\rule{1em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}|-40|\phantom{\rule{1em}{0ex}}$
3. $\phantom{\rule{0.2em}{0ex}}|22|$

In the following exercises, evaluate each absolute value expression.

1. $\phantom{\rule{0.2em}{0ex}}|x|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=-28$
2. $\phantom{\rule{0.2em}{0ex}}|-u|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}u=-15$
1. 28
2. 15
1. $\phantom{\rule{0.2em}{0ex}}|y|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}y=-37$
2. $\phantom{\rule{0.2em}{0ex}}|-z|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}z=-24$
1. $\phantom{\rule{0.2em}{0ex}}-|p|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}p=19$
2. $\phantom{\rule{0.2em}{0ex}}-|q|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}q=-33$
1. −19
2. −33
1. $\phantom{\rule{0.2em}{0ex}}-|a|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}a=60$
2. $\phantom{\rule{0.2em}{0ex}}-|b|\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}b=-12$

In the following exercises, fill in $\text{<},\text{>},\text{or}=$ to compare each expression.

1. $\phantom{\rule{0.2em}{0ex}}-6\text{__}|-6|\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-|-3|\text{__}-3$
1. <
2. =
1. $\phantom{\rule{0.2em}{0ex}}-8\text{__}|-8|\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}-|-2|\text{__}-2$
1. $\phantom{\rule{0.2em}{0ex}}|-3|\text{__}-|-3|\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}4\text{__}-|-4|$
1. >
2. >
1. $\phantom{\rule{0.2em}{0ex}}|-5|\text{__}-|-5|\phantom{\rule{0.2em}{0ex}}$
2. $\phantom{\rule{0.2em}{0ex}}9\text{__}-|-9|$

In the following exercises, simplify each expression.

$|8-4|$

4

$|9-6|$

$8|-7|$

56

$5|-5|$

$|15-7|-|14-6|$

0

$|17-8|-|13-4|$

$18-|2\left(8-3\right)|$

8

$15-|3\left(8-5\right)|$

$8\left(14-2|-2|\right)$

80

$6\left(13-4|-2|\right)$

Translate Word Phrases into Expressions with Integers

Translate each phrase into an expression with integers. Do not simplify .

1. the opposite of $8$
2. the opposite of $-6$
3. negative three
4. $4$ minus negative $3$
1. −8
2. −(−6), or 6
3. −3
4. 4−(−3)
1. the opposite of $11$
2. the opposite of $-4$
3. negative nine
4. $8$ minus negative $2$
1. the opposite of $20$
2. the opposite of $-5$
3. negative twelve
4. $18$ minus negative $7$
1. −20
2. −(−5), or 5
3. −12
4. 18−(−7)
1. the opposite of $15$
2. the opposite of $-9$
3. negative sixty
4. $\phantom{\rule{0.2em}{0ex}}12$ minus $5$

a temperature of $6\phantom{\rule{0.2em}{0ex}}\text{degrees}$ below zero

−6 degrees

a temperature of $14\phantom{\rule{0.2em}{0ex}}\text{degrees}$ below zero

an elevation of $40\phantom{\rule{0.2em}{0ex}}\text{feet}$ below sea level

−40 feet

an elevation of $65\phantom{\rule{0.2em}{0ex}}\text{feet}$ below sea level

a football play loss of $12\phantom{\rule{0.2em}{0ex}}\text{yards}$

−12 yards

a football play gain of $4\phantom{\rule{0.2em}{0ex}}\text{yards}$

a stock gain of $\text{3}$

$3 a stock loss of $\text{5}$ a golf score one above par +1 a golf score of $3$ below par Everyday math Elevation The highest elevation in the United States is Mount McKinley, Alaska, at $20,320\phantom{\rule{0.2em}{0ex}}\text{feet}$ above sea level. The lowest elevation is Death Valley, California, at $282\phantom{\rule{0.2em}{0ex}}\text{feet}$ below sea level. Use integers to write the elevation of: 1. Mount McKinley 2. Death Valley 1. 20,320 feet 2. −282 feet Extreme temperatures The highest recorded temperature on Earth is $\text{58° Celsius,}$ recorded in the Sahara Desert in 1922. The lowest recorded temperature is $\text{90°}$ below $\text{0° Celsius,}$ recorded in Antarctica in 1983. Use integers to write the: 1. highest recorded temperature 2. lowest recorded temperature State budgets In June, 2011, the state of Pennsylvania estimated it would have a budget surplus of $\text{540 million.}$ That same month, Texas estimated it would have a budget deficit of $\text{27 billion.}$ Use integers to write the budget: 1. surplus 2. deficit 1.$540 million
2. −\$27 billion

College enrollments Across the United States, community college enrollment grew by $1,400,000$ students from $2007$ to $2010.$ In California, community college enrollment declined by $110,171$ students from $2009$ to $2010.$ Use integers to write the change in enrollment:

1. growth
2. decline

Writing exercises

Give an example of a negative number from your life experience.

Sample answer: I have experienced negative temperatures.

What are the three uses of the “−” sign in algebra? Explain how they differ.

Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
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a perfect square v²+2v+_
kkk nice
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Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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yes
Asali
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Asali
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China
Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
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