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Introduction to Integers

Locate positive and negative numbers on the number line

Locate Positive and Negative Numbers on the Number Line

Do you live in a place that has very cold winters? Have you ever experienced a temperature below zero? If so, you are already familiar with negative numbers. A negative number is a number that is less than $0.$ Very cold temperatures are measured in degrees below zero and can be described by negative numbers. For example, $-1\text{^{\circ}F}$ (read as “negative one degree Fahrenheit”) is $1\ \text{degree}$ below $0.$ A minus sign is shown before a number to indicate that it is negative. shows $-20\text{^{\circ}F},$ which is $20\ \text{degrees}$ below $0.$

Temperatures are not the only negative numbers. A bank overdraft is another example of a negative number. If a person writes a check for more than he has in his account, his balance will be negative.

Elevations can also be represented by negative numbers. The elevation at sea level is $\text{0 feet}.$ Elevations above sea level are positive and elevations below sea level are negative. The elevation of the Dead Sea, which borders Israel and Jordan, is about $1,302\ \text{feet}$ below sea level, so the elevation of the Dead Sea can be represented as $-1,302\ \text{feet}.$ See .

Depths below the ocean surface are also described by negative numbers. A submarine, for example, might descend to a depth of $500\ \text{feet}.$ Its position would then be $-500\ \text{feet}$ as labeled in .

Both positive and negative numbers can be represented on a number line. Recall that the number line created in Add Whole Numbers started at $0$ and showed the counting numbers increasing to the right as shown in . The counting numbers $\text{(1, 2, 3, \ldots )}$ on the number line are all positive. We could write a plus sign, $+,$ before a positive number such as $+2$ or $+3,$ but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.

Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with $-1$ at the first mark to the left of $0,-2$ at the next mark, and so on. See .

The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number.

Condensed — the full section is in OpenStax Prealgebra 2e.

Order Positive and Negative Numbers

We can use the number line to compare and order positive and negative numbers. Going from left to right, numbers increase in value. Going from right to left, numbers decrease in value. See .

Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers. Remember that we use the notation $ais less than \(b$) when $a$ is to the left of $b$ on the number line. We write $a>b$ (read $a$ is greater than $b$) when $a$ is to the right of $b$ on the number line. This is shown for the numbers $3$ and $5$ in .

The numbers lines to follow show a few more examples.

ⓐ

$4$ is to the right of $1$ on the number line, so $4>1.$

$1$ is to the left of $4$ on the number line, so $1<4.$

ⓑ

Example

Try it.

Order each of the following pairs of numbers using $<$ or $\text{>:}$

ⓐ $\ 14___6\$
ⓑ $\ -1___9\$
ⓒ $\ -1___-4\$
ⓓ $\ 2___-20$

Solution

Begin by plotting the numbers on a number line as shown in .

ⓐ Compare 14 and 6.	$14___6$
14 is to the right of 6 on the number line.	$14>6$

ⓑ Compare −1 and 9.	$-1___9$
−1 is to the left of 9 on the number line.	$-1<9$

ⓒ Compare −1 and −4.	$-1___-4$
−1 is to the right of −4 on the number line.	$-1>-4$

ⓓ Compare 2 and −20.	$2___-20$
2 is to the right of −20 on the number line.	$2>-20$

Condensed — the full section is in OpenStax Prealgebra 2e.

Find Opposites

On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers $2$ and $-2$ are the same distance from zero, they are called opposites. The opposite of $2$ is $-2,$ and the opposite of $-2$ is $2$ as shown in (a). Similarly, $3$ and $-3$ are opposites as shown in (b).

Example

Try it.

Find the opposite of each number:

ⓐ $\ 7$
ⓑ $\ -10$

Solution

ⓐ The number $-7$ is the same distance from $0$ as $7,$ but on the opposite side of $0.$ So $-7$ is the opposite of $7$ as shown in .
ⓑ The number $10$ is the same distance from $0$ as $-10$, but on the opposite side of $0.$ So $10$ is the opposite of $-10$ as shown in .

Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol $\text{“-”,}$ in three different ways.

$10-4$	Between two numbers, the symbol indicates the operation of subtraction. We read $10-4$ as 10 minus $4$.
$-8$	In front of a number, the symbol indicates a negative number. We read $-8$ as negative eight.
$-x$	In front of a variable or a number, it indicates the opposite. We read$-x$ as the opposite of $x$.
$-(-2)$	Here we have two signs. The sign in the parentheses indicates that the number is negative 2. The sign outside the parentheses indicates the opposite. We read $-(-2)$ as the opposite of $-2.$

Example

Try it.

Simplify: $-(-6).$

Solution

	$-(-6)$
The opposite of $-6$ is $6.$	$6$

The set of counting numbers, their opposites, and $0$ is the set of integers.

We must be very careful with the signs when evaluating the opposite of a variable.

Example

Try it.

Evaluate $-x:$

ⓐ when $x=8$
ⓑ when $x=-8.$

Solution

ⓐ To evaluate $-x$ when $x=8$, substitute $8$ for $x$.
	$-x$

Simplify.	$-8$

ⓑ To evaluate $-x$ when $x=-8$, substitute $-8$ for $x$.
	$-x$

Simplify.	$8$

Simplify Expressions with Absolute Value

We saw that numbers such as $5$ and $-5$ are opposites because they are the same distance from $0$ on the number line. They are both five units from $0.$ The distance between $0$ and any number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative.

The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of $5$ is written as $|5|,$ and the absolute value of $-5$ is written as $|-5|$ as shown in .

Example

Try it.

Simplify:

ⓐ $\ |3|$
ⓑ $\ |-44|$
ⓒ $\ |0|$

Solution

ⓐ
	$\|3\|$
3 is 3 units from zero.	$3$

ⓑ
	$\|-44\|$
−44 is 44 units from zero.	$44$

ⓒ
	$\|0\|$
0 is already at zero.	$0$

We treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first.

Example

Try it.

Evaluate:

ⓐ $\ |x|\ \text{when}\ x=-35$
ⓑ $\ |\text{-y}|\ \text{when}\ y=-20$
ⓒ $\ -|u|\ \text{when}\ u=12$
ⓓ $\ -|p|\ \text{when}\ p=-14$

Solution

ⓐ To find $\|x\|$ when $x=-35:$
	$\|x\|$

Take the absolute value.	$35$

ⓑ To find $\|-y\|$ when $y=-20:$
	$\|-y\|$

Simplify.	$\|20\|$
Take the absolute value.	$20$

ⓒ To find $-\|u\|$ when $u=12:$
	$-\|u\|$

Take the absolute value.	$-12$

ⓓ To find $-\|p\|$ when $p=-14:$
	$-\|p\|$

Take the absolute value.	$-14$

Notice that the result is negative only when there is a negative sign outside the absolute value symbol.

Example

Try it.

Fill in $\text{<},\text{>},\text{or}=$ for each of the following:

ⓐ $\ |-5|___-|-5|\$
ⓑ $\ 8___-|-8|\$
ⓒ $\ -9___-|-9|\$
ⓓ $\ -|-7|___-7$

Solution

To compare two expressions, simplify each one first. Then compare.

ⓐ
	$\|-5\|___-\|-5\|$
Simplify.	$5___-5$
Order.	$5>-5$

ⓑ
	$8___-\|-8\|$
Simplify.	$8___-8$
Order.	$8>-8$

ⓒ
	$-9___-\|-9\|\$
Simplify.	$-9___-9$
Order.	$-9=-9$

ⓓ
	$-\|-7\|___-7$
Simplify.	$-7___-7$
Order.	$-7=-7$

Condensed — the full section is in OpenStax Prealgebra 2e.

Translate Word Phrases into Expressions with Integers

Now we can translate word phrases into expressions with integers. Look for words that indicate a negative sign. For example, the word negative in “negative twenty” indicates $-20.$ So does the word opposite in “the opposite of $20\text{.”}$

Example

Try it.

Translate each phrase into an expression with integers:

ⓐ the opposite of positive fourteen
ⓑ the opposite of $-11$
ⓒ negative sixteen
ⓓ two minus negative seven

Solution

ⓐ the opposite of fourteen
$-14$
ⓑ the opposite of −11
$-(-11)=11$
ⓒ negative sixteen
$-16$
ⓓ two minus negative seven
$2-(-7)$

As we saw at the start of this section, negative numbers are needed to describe many real-world situations. We’ll look at some more applications of negative numbers in the next example.

Example

Try it.

Translate into an expression with integers:

ⓐ The temperature is $12\ \text{degrees Fahrenheit}$ below zero.
ⓑ The football team had a gain of $3\ \text{yards.}$
ⓒ The elevation of the Dead Sea is $1,302\ \text{feet}$ below sea level.
ⓓ A checking account is overdrawn by $\text{\$40.}$

Solution

Look for key phrases in each sentence. Then look for words that indicate negative signs. Don’t forget to include units of measurement described in the sentence.

ⓐ	The temperature is 12 degrees Fahrenheit below zero.
Below zero tells us that 12 is a negative number.	$-12ºF$

ⓑ	The football team had a gain of 3 yards.
A gain tells us that 3 is a positive number.	$3$ yards

ⓒ	The elevation of the Dead Sea is 1,302 feet below sea level.
Below sea level tells us that 1,302 is a negative number.	$-1,302$ feet

ⓓ	A checking account is overdrawn by $40.
Overdrawn tells us that 40 is a negative number.	$-\$40$

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.

Introduction to Integers

Locate Positive and Negative Numbers on the Number Line

For the following exercises, draw a number line and locate and label the given points on that number line.

Try it.

ⓐ $\ 2\$
ⓑ $\ -2\$
ⓒ $\ -5$

Solution

Try it.

ⓐ $\ 5\$
ⓑ $\ -5\$
ⓒ $\ -2$

Try it.

ⓐ $\ -8\$
ⓑ $\ 8\$
ⓒ $\ -6$

Solution

Try it.

ⓐ $\ -7\$
ⓑ $\ 7\$
ⓒ $\ -1$

Order Positive and Negative Numbers on the Number Line

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

Try it.

ⓐ $\ 9\text{__}4\$
ⓑ $\ -3\text{__}6\$
ⓒ $\ -8\text{__}-2\$
ⓓ $\ 1\text{__}-10$

Solution

ⓐ >
ⓑ <
ⓒ <
ⓓ >

Try it.

ⓐ $\ 6\text{__}2;\$
ⓑ $\ -7\text{__}4;\$
ⓒ $\ -9\text{__}-1;\$
ⓓ $\ 9\text{__}-3$

Try it.

ⓐ $\ -5\text{__}1;\$
ⓑ $\ -4\text{__}-9;\$
ⓒ $\ 6\text{__}10;\$
ⓓ $\ 3\text{__}-8$

Solution

ⓐ <
ⓑ >
ⓒ <
ⓓ >

Try it.

ⓐ $\ -7\text{__}3;\$
ⓑ $\ -10\text{__}-5;\$
ⓒ $\ 2\text{__}-6;\$
ⓓ $\ 8\text{__}9$

Find Opposites

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

Try it.

ⓐ $\ 2\$
ⓑ $\ -6$

Solution

ⓐ −2
ⓑ 6

Try it.

ⓐ $\ 9\$
ⓑ $\ -4$

Try it.

ⓐ $\ -8\$
ⓑ $\ 1$

Solution

ⓐ 8
ⓑ −1

Try it.

ⓐ $\ -2\$
ⓑ $\ 6$

In the following exercises, simplify.

Try it.

$-(-4)$

Solution

Try it.

$-(-8)$

Try it.

$-(-15)$

Solution

Try it.

$-(-11)$

In the following exercises, evaluate.

Try it.

$-m\ \text{when}\$

ⓐ $\ m=3\$
ⓑ $\ m=-3$

Solution

ⓐ −3
ⓑ 3

Try it.

$-p\ \text{when}\$

ⓐ $\ p=6\$
ⓑ $\ p=-6$

Try it.

$-c\ \text{when}\$

ⓐ $\ c=12\$
ⓑ $\ c=-12$

Solution

ⓐ −12;
ⓑ 12

Try it.

$-d\ \text{when}\$

ⓐ $\ d=21\$
ⓑ $\ d=-21$

Simplify Expressions with Absolute Value

In the following exercises, simplify each absolute value expression.

Try it.

ⓐ $\ |7|\$
ⓑ $\ |-25|\$
ⓒ$\ |0|$

Solution

ⓐ 7
ⓑ 25
ⓒ 0

Try it.

ⓐ $\ |5|\$
ⓑ $\ |20|\$
ⓒ $\ |-19|$

Try it.

ⓐ $\ |-32|\$
ⓑ $\ |-18|\$
ⓒ $\ |16|$

Solution

ⓐ 32
ⓑ 18
ⓒ 16

Try it.

ⓐ $\ |-41|\$
ⓑ $\ |-40|\$
ⓒ $\ |22|$

In the following exercises, evaluate each absolute value expression.

Try it.

ⓐ $\ |x|\ \text{when}\ x=-28$
ⓑ $\ |-u|\ \text{when}\ u=-15$

Solution

ⓐ 28
ⓑ 15

Try it.

ⓐ $\ |y|\ \text{when}\ y=-37$
ⓑ $\ |-z|\ \text{when}\ z=-24$

Try it.

ⓐ $\ -|p|\ \text{when}\ p=19$
ⓑ $\ -|q|\ \text{when}\ q=-33$

Solution

ⓐ −19
ⓑ −33

Try it.

ⓐ $\ -|a|\ \text{when}\ a=60$
ⓑ $\ -|b|\ \text{when}\ b=-12$

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

Try it.

ⓐ $\ -6\text{__}|-6|\$
ⓑ $\ -|-3|\text{__}-3$

Solution

ⓐ <
ⓑ =

Try it.

ⓐ $\ -8\text{__}|-8|\$
ⓑ $\ -|-2|\text{__}-2$

Try it.

ⓐ $\ |-3|\text{__}-|-3|\$
ⓑ $\ 4\text{__}-|-4|$

Solution

ⓐ >
ⓑ >

Try it.

ⓐ $\ |-5|\text{__}-|-5|\$
ⓑ$\ 9\text{__}-|-9|$

In the following exercises, simplify each expression.

Try it.

$|8-4|$

Solution

Try it.

$|9-6|$

Try it.

$8|-7|$

Solution

Try it.

$5|-5|$

Try it.

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

Solution

Try it.

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

Try it.

$18-|2(8-3)|$

Solution

Try it.

$15-|3(8-5)|$

Try it.

$8(14-2|-2|)$

Solution

Try it.

$6(13-4|-2|)$

Translate Word Phrases into Expressions with Integers

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

Try it.

ⓐ the opposite of $8$
ⓑ the opposite of $-6$
ⓓ $4$ minus negative $3$

Solution

ⓐ −8
ⓑ −(−6), or 6
ⓒ −3
ⓓ 4−(−3)

Try it.

ⓐ the opposite of $11$
ⓑ the opposite of $-4$
ⓓ $8$ minus negative $2$

Try it.

ⓐ the opposite of $20$
ⓑ the opposite of $-5$
ⓓ $18$ minus negative $7$

Solution

ⓐ −20
ⓑ −(−5), or 5
ⓒ −12
ⓓ 18−(−7)

Try it.

ⓐ the opposite of $15$
ⓑ the opposite of $-9$
ⓓ $\ 12$ minus $5$

Try it.

a temperature of $6\ \text{degrees}$ below zero

Solution

−6 degrees

Try it.

a temperature of $14\ \text{degrees}$ below zero

Try it.

an elevation of $40\ \text{feet}$ below sea level

Solution

−40 feet

Try it.

an elevation of $65\ \text{feet}$ below sea level

Try it.

a football play loss of $12\ \text{yards}$

Solution

−12 yards

Try it.

a football play gain of $4\ \text{yards}$

Try it.

a stock gain of $\text{\$3}$

Solution

Try it.

a stock loss of $\text{\$5}$

Try it.

a golf score one above par

Solution

Try it.

a golf score of $3$ below par

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Plot $0,1,\text{and}\ 3$ on a number line.
If you missed this problem, review .

답을 드러내세요
Fill in the appropriate symbol: $\text{(=, <, or >):}\ 2___4$
If you missed this problem, review .

답을 드러내세요

$<$
Plot the numbers on a number line:
1. ⓐ $\ 3\$
2. ⓑ $\ -3\$
3. ⓒ $\ -2$
답을 드러내세요
Draw a number line. Mark $0$ in the center and label several units to the left and right.
1. ⓐ To plot $3,$ start at $0$ and count three units to the right. Place a point as shown in .
2. ⓑ To plot $-3,$ start at $0$ and count three units to the left. Place a point as shown in .
3. ⓒ To plot $-2,$ start at $0$ and count two units to the left. Place a point as shown in .
Plot the numbers on a number line.
1. ⓐ $\ -4\$
2. ⓑ $\ 4\$
3. ⓐ $\ -1$
답을 드러내세요
Order each of the following pairs of numbers using $<$ or $\text{>:}$
1. ⓐ $\ 14___6\$
2. ⓑ $\ -1___9\$
3. ⓒ $\ -1___-4\$
4. ⓓ $\ 2___-20$
답을 드러내세요

Begin by plotting the numbers on a number line as shown in .

ⓐ Compare 14 and 6. $14___6$
14 is to the right of 6 on the number line. $14>6$
ⓑ Compare −1 and 9. $-1___9$
−1 is to the left of 9 on the number line. $-1<9$
ⓒ Compare −1 and −4. $-1___-4$
−1 is to the right of −4 on the number line. $-1>-4$
ⓓ Compare 2 and −20. $2___-20$
2 is to the right of −20 on the number line. $2>-20$
Order each of the following pairs of numbers using $<$ or $\text{>.}$
1. ⓐ $\ 15___7\$
2. ⓑ $\ -2___5\$
3. ⓒ $\ -3___-7\$
4. ⓓ $\ 5___-17$
답을 드러내세요
1. ⓐ >
2. ⓑ <
3. ⓒ >
4. ⓓ >
Order each of the following pairs of numbers using $<$ or $\text{>.}$
1. ⓐ $\ 8___13\$
2. ⓑ $\ 3___-4\$
3. ⓒ $\ -5___-2\$
4. ⓓ $\ 9___-21$
답을 드러내세요
1. ⓐ <
2. ⓑ >
3. ⓒ <
4. ⓓ >
Find the opposite of each number:
1. ⓐ $\ 7$
2. ⓑ $\ -10$
답을 드러내세요
1. ⓐ The number $-7$ is the same distance from $0$ as $7,$ but on the opposite side of $0.$ So $-7$ is the opposite of $7$ as shown in .
2. ⓑ The number $10$ is the same distance from $0$ as $-10$, but on the opposite side of $0.$ So $10$ is the opposite of $-10$ as shown in .
Find the opposite of each number:
1. ⓐ $\ 4\$
2. ⓑ $\ -3$
답을 드러내세요
1. ⓐ −4
2. ⓑ 3
Find the opposite of each number:
1. ⓐ $\ 8\$
2. ⓑ $\ -5$
답을 드러내세요
1. ⓐ −8
2. ⓑ 5
Simplify: $-(-6).$

답을 드러내세요

$-(-6)$
The opposite of $-6$ is $6.$ $6$
Simplify:
$-(-1)$

답을 드러내세요

1
Simplify:
$-(-5)$

답을 드러내세요

5
Evaluate $-x:$
1. ⓐ when $x=8$
2. ⓑ when $x=-8.$
답을 드러내세요

ⓐ To evaluate $-x$ when $x=8$, substitute $8$ for $x$.
$-x$
Simplify. $-8$
ⓑ To evaluate $-x$ when $x=-8$, substitute $-8$ for $x$.
$-x$
Simplify. $8$
Evaluate $-n:\$
1. ⓐ $\ \text{when}\ n=4\$
2. ⓑ $\ \text{when}\ n=-4$
답을 드러내세요
1. ⓐ −4
2. ⓑ 4
Evaluate: $-m:\$
1. ⓐ $\ \text{when}\ m=11\$
2. ⓑ $\ \text{when}\ m=-11$
답을 드러내세요
1. ⓐ −11
2. ⓑ 11
Simplify:
1. ⓐ $\ |3|$
2. ⓑ $\ |-44|$
3. ⓒ $\ |0|$
답을 드러내세요

ⓐ
$|3|$
3 is 3 units from zero. $3$
ⓑ
$|-44|$
−44 is 44 units from zero. $44$
ⓒ
$|0|$
0 is already at zero. $0$
Simplify:
1. ⓐ $\ |12|\$
2. ⓑ $\ -|-28|$
답을 드러내세요
1. ⓐ 12
2. ⓑ −28
Simplify:
1. ⓐ $\ |9|\$
2. ⓑ $-|37|$
답을 드러내세요
1. ⓐ 9
2. ⓑ −37
Evaluate:
1. ⓐ $\ |x|\ \text{when}\ x=-35$
2. ⓑ $\ |\text{-y}|\ \text{when}\ y=-20$
3. ⓒ $\ -|u|\ \text{when}\ u=12$
4. ⓓ $\ -|p|\ \text{when}\ p=-14$
답을 드러내세요

ⓐ To find $|x|$ when $x=-35:$
$|x|$
Take the absolute value. $35$
ⓑ To find $|-y|$ when $y=-20:$
$|-y|$
Simplify. $|20|$
Take the absolute value. $20$
ⓒ To find $-|u|$ when $u=12:$
$-|u|$
Take the absolute value. $-12$
ⓓ To find $-|p|$ when $p=-14:$
$-|p|$
Take the absolute value. $-14$
Notice that the result is negative only when there is a negative sign outside the absolute value symbol.
Evaluate:
1. ⓐ $\ |x|\ \text{when}\ x=-17\$
2. ⓑ $\ |\text{-y}|\ \text{when}\ y=-39\$
3. ⓒ $\ -|m|\ \text{when}\ m=22\$
4. ⓓ $\ -|p|\ \text{when}\ p=-11$
답을 드러내세요
1. ⓐ 17
2. ⓑ 39
3. ⓒ −22
4. ⓓ −11
1. ⓐ $\ |y|\ \text{when}\ y=-23\$
2. ⓑ $\ |-y|\ \text{when}\ y=-21$
3. ⓒ $\ -|n|\ \text{when}\ n=37\$
4. ⓓ $\ -|q|\ \text{when}\ q=-49$
답을 드러내세요
1. ⓐ 23
2. ⓑ 21
3. ⓒ −37
4. ⓓ −49
Fill in $\text{<},\text{>},\text{or}=$ for each of the following:
1. ⓐ $\ |-5|___-|-5|\$
2. ⓑ $\ 8___-|-8|\$
3. ⓒ $\ -9___-|-9|\$
4. ⓓ $\ -|-7|___-7$
답을 드러내세요

To compare two expressions, simplify each one first. Then compare.
ⓐ
$|-5|___-|-5|$
Simplify. $5___-5$
Order. $5>-5$
ⓑ
$8___-|-8|$
Simplify. $8___-8$
Order. $8>-8$
ⓒ
$-9___-|-9|\$
Simplify. $-9___-9$
Order. $-9=-9$
ⓓ
$-|-7|___-7$
Simplify. $-7___-7$
Order. $-7=-7$
Fill in $\text{<},\text{>},\text{or}=\ \text{for each of the following:}$
1. ⓐ $\ |-9|\ \text{___}-|-9|\$
2. ⓑ $\ 2___-|-2|\$
3. ⓒ $\ -8___|-8|\$
4. ⓓ $\ -|-5|\text{___}-5$
답을 드러내세요
1. ⓐ >
2. ⓑ >
3. ⓒ <
4. ⓓ =
Fill in $\text{<},\text{>},\text{or}=$ for each of the following:
1. ⓐ $\ 7\text{___}-|-7|\$
2. ⓑ $\ -|-11|\text{___}-11\$
3. ⓒ $\ |-4|\text{___}-|-4|\$
4. ⓓ $\ -1\text{___}|-1|$
답을 드러내세요
1. ⓐ >
2. ⓑ =
3. ⓒ >
4. ⓓ <
Simplify:
1. ⓐ $\ |9-3|\ \$
2. ⓑ $\ 4|-2|$
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For each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.
ⓐ
|9−3|
Simplify inside the absolute value sign. |6|
Take the absolute value. 6
ⓑ
4|−2|
Take the absolute value. 4⋅2
Multiply. 8
Simplify:
1. ⓐ $\ |12-9|\$
2. ⓑ $\ 3|-6|$
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1. ⓐ 3
2. ⓑ 18
Simplify:
1. ⓐ $\ |27-16|\$
2. ⓑ $\ 9|-7|$
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1. ⓐ 11
2. ⓑ 63
Simplify: $|8+7|-|5+6|.$

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For each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses.
|8+7|−|5+6|
Simplify inside each absolute value sign. |15|−|11|
Subtract. 4
Simplify: $|1+8|-|2+5|$

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2
Simplify: $|9-5|-|7-6|$

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3
Simplify: $24-|19-3(6-2)|.$

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We use the order of operations. Remember to simplify grouping symbols first, so parentheses inside absolute value symbols would be first.
$24-|19-3(6-2)|$
Simplify in the parentheses first. $24-|19-3(4)|$
Multiply $3(4)$. $24-|19-12|$
Subtract inside the absolute value sign. $24-|7|$
Take the absolute value. $24-7$
Subtract. $17$
Simplify: $19-|11-4(3-1)|$

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16
Simplify: $9-|8-4(7-5)|$

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9
Translate each phrase into an expression with integers:
1. ⓐ the opposite of positive fourteen
2. ⓑ the opposite of $-11$
3. ⓒ negative sixteen
4. ⓓ two minus negative seven
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1. ⓐ the opposite of fourteen
  $-14$
2. ⓑ the opposite of −11
  $-(-11)=11$
3. ⓒ negative sixteen
  $-16$
4. ⓓ two minus negative seven
  $2-(-7)$
Translate each phrase into an expression with integers:
1. ⓐ the opposite of positive nine
2. ⓑ the opposite of $-15$
3. ⓒ negative twenty
4. ⓓ eleven minus negative four
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1. ⓐ −9
2. ⓑ 15
3. ⓒ −20
4. ⓓ 11−(−4)
Translate each phrase into an expression with integers:
1. ⓐ the opposite of negative nineteen
2. ⓑ the opposite of twenty-two
3. ⓒ negative nine
4. ⓓ negative eight minus negative five
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1. ⓐ 19
2. ⓑ −22
3. ⓒ −9
4. ⓓ −8−(−5)
Translate into an expression with integers:
1. ⓐ The temperature is $12\ \text{degrees Fahrenheit}$ below zero.
2. ⓑ The football team had a gain of $3\ \text{yards.}$
3. ⓒ The elevation of the Dead Sea is $1,302\ \text{feet}$ below sea level.
4. ⓓ A checking account is overdrawn by $\text{\$40.}$
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Look for key phrases in each sentence. Then look for words that indicate negative signs. Don’t forget to include units of measurement described in the sentence.
ⓐ The temperature is 12 degrees Fahrenheit below zero.
Below zero tells us that 12 is a negative number. $-12ºF$
ⓑ The football team had a gain of 3 yards.
A gain tells us that 3 is a positive number. $3$ yards
ⓒ The elevation of the Dead Sea is 1,302 feet below sea level.
Below sea level tells us that 1,302 is a negative number. $-1,302$ feet
ⓓ A checking account is overdrawn by $40.
Overdrawn tells us that 40 is a negative number. $-\$40$
Translate into an expression with integers:
The football team had a gain of $5\ \text{yards.}$

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5 yards

Symbols used here

^\circ

degrees
1/360 of a full turn. 180° = π radians.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Introduction to Integers

Locate positive and negative numbers on the number line
Order positive and negative numbers
Find opposites
Simplify expressions with absolute value
Translate word phrases to expressions with integers
ⓐ To plot
ⓑ To plot
ⓒ To plot

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

자신만의 길을 찾아보세요

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

에 더 Arithmetic

Order of operations Fractions Percentages Powers and roots

\(10-4\)	Between two numbers, the symbol indicates the operation of subtraction. We read \(10-4\) as 10 minus \(4\).
\(-8\)	In front of a number, the symbol indicates a negative number. We read \(-8\) as negative eight.
\(-x\)	In front of a variable or a number, it indicates the opposite. We read\(-x\) as the opposite of \(x\).
\(-(-2)\)	Here we have two signs. The sign in the parentheses indicates that the number is negative 2. The sign outside the parentheses indicates the opposite. We read \(-(-2)\) as the opposite of \(-2.\)

ⓐ To evaluate \(-x\) when \(x=8\), substitute \(8\) for \(x\).
	\(-x\)

Simplify.	\(-8\)

ⓑ To evaluate \(-x\) when \(x=-8\), substitute \(-8\) for \(x\).
	\(-x\)

Simplify.	\(8\)

ⓐ To find \(\|x\|\) when \(x=-35:\)
	\(\|x\|\)

Take the absolute value.	\(35\)

ⓑ To find \(\|-y\|\) when \(y=-20:\)
	\(\|-y\|\)

Simplify.	\(\|20\|\)
Take the absolute value.	\(20\)

ⓐ Compare 14 and 6.	\(14___6\)
14 is to the right of 6 on the number line.	\(14>6\)

ⓑ Compare −1 and 9.	\(-1___9\)
−1 is to the left of 9 on the number line.	\(-1<9\)

ⓒ Compare −1 and −4.	\(-1___-4\)
−1 is to the right of −4 on the number line.	\(-1>-4\)

ⓓ Compare 2 and −20.	\(2___-20\)
2 is to the right of −20 on the number line.	\(2>-20\)

ⓒ To find \(-\|u\|\) when \(u=12:\)
	\(-\|u\|\)

Take the absolute value.	\(-12\)

ⓓ To find \(-\|p\|\) when \(p=-14:\)
	\(-\|p\|\)

Take the absolute value.	\(-14\)

ⓐ	The temperature is 12 degrees Fahrenheit below zero.
Below zero tells us that 12 is a negative number.	\(-12ºF\)

ⓑ	The football team had a gain of 3 yards.
A gain tells us that 3 is a positive number.	\(3\) yards

ⓒ	The elevation of the Dead Sea is 1,302 feet below sea level.
Below sea level tells us that 1,302 is a negative number.	\(-1,302\) feet

ⓓ	A checking account is overdrawn by $40.
Overdrawn tells us that 40 is a negative number.	\(-\$40\)

	\(-(-6)\)
The opposite of \(-6\) is \(6.\)	\(6\)

ⓐ
	\(\|-5\|___-\|-5\|\)
Simplify.	\(5___-5\)
Order.	\(5>-5\)

ⓑ
	\(8___-\|-8\|\)
Simplify.	\(8___-8\)
Order.	\(8>-8\)

ⓒ
	\(-9___-\|-9\|\\)
Simplify.	\(-9___-9\)
Order.	\(-9=-9\)

ⓐ
	\|9−3\|
Simplify inside the absolute value sign.	\|6\|
Take the absolute value.	6

	\|8+7\|−\|5+6\|
Simplify inside each absolute value sign.	\|15\|−\|11\|
Subtract.	4

	\(24-\|19-3(6-2)\|\)
Simplify in the parentheses first.	\(24-\|19-3(4)\|\)
Multiply \(3(4)\).	\(24-\|19-12\|\)
Subtract inside the absolute value sign.	\(24-\|7\|\)
Take the absolute value.	\(24-7\)
Subtract.	\(17\)