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Integers

Simplify expressions with absolute value

Simplify Expressions with Absolute Value

A negative number is a number less than 0. The negative numbers are to the left of zero on the number line. See .

You may have noticed that, 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, each one is called the opposite of the other. The opposite of $2$ is $-2,$ and the opposite of $-2$ is $2.$

illustrates the definition.

We saw that numbers such as 3 and $-3$ are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

For example,

\[\begin{array}{l}-5\ \text{is 5 units away from 0, so}\ |-5|=5. \\ \text{5 is 5 units away from 0, so}\ |5|=5.\end{array}\]

illustrates this idea.

The absolute value of a number is never negative because distance cannot be negative. The only number with absolute value equal to zero is the number zero itself because the distance from 0 to 0 on the number line is zero units.

Example

Try it.

Fill in $<,>,$ or $=$ for each of the following pairs of numbers:

ⓐ $|-5|__-|-5|$ ⓑ $8__-|-8|$ ⓒ $-9__-|-9|$ ⓓ $\text{-}(-16)__|-16|.$

Solution

ⓐ

	$\|-5\|__-\|-5\|$
Simplify.	$\ 5__-5$
Order.	$\ 5>-5$
	$\|-5\|>-\|-5\|$

ⓑ

	$8__-\|-8\|$
Simplify.	$8__-8$
Order.	$8>-8$
	$8>-\|-8\|$

ⓒ

	$-9__-\|-9\|$
Simplify.	$-9__-9$
Order.	$-9=-9$
	$-9=-\|-9\|$

ⓓ

	$-(-16)__\|-16\|$
Simplify.	$\ 16__16$
Order.	$\ 16=16$
	$-(-16)=\|-16\|$

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Add and Subtract Integers

So far, we have only used the counting numbers and the whole numbers.

\[\begin{array}{llll}\text{Counting numbers} & & & 1,2,3\ldots \\ \text{Whole numbers} & & & 0,1,2,3\ldots .\end{array}\]

Our work with opposites gives us a way to define the integers. The whole numbers and their opposites are called the integers. The integers are the numbers $\ldots -3,-2,-1,0,1,2,3\text{\ldots }$

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.

We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.

We let one color (blue) represent positive. The other color (red) will represent the negatives.

If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.

We will use the counters to show how to add:

\[5+3\ -5+(-3)\ -5+3\ 5+(-3)\]

Example

Try it.

Add: ⓐ $-1+(-4)$ ⓑ $-1+5$ ⓒ $1+(-5).$

Solution

ⓐ



1 negative plus 4 negatives is 5 negatives

ⓑ



There are more positives, so the sum is positive.

ⓒ



There are more negatives, so the sum is negative.

\[5-3\ -5-(-3)\ -5-3\ 5-(-3)\]


Model the first number.
We now add the needed neutral pairs.
We remove the number of counters modeled by the second number.
Count what is left.

Example

Try it.

Subtract: ⓐ $3-1$ ⓑ $-3-(-1)$ ⓒ $-3-1$ ⓓ $3-(-1).$

Solution

ⓐ

	$\$
Take 1 negative from 3 negatives and get 2 negatives positives.

ⓑ


Take 1 positive from 3 negatives and get 2 negatives.

ⓒ

	$\$
Take 1 positive from the one added neutral pair.

ⓓ

	$\$
Take 1 negative from the one added neutral pair.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Multiply and Divide Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we are using the model just to help us discover the pattern.

We remember that $a\cdot b$ means add a, b times.

The next two examples are more interesting. What does it mean to multiply 5 by $-3?$ It means subtract $5,3$ times. Looking at subtraction as “taking away”, it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace.

In summary:

\[\begin{array}{llll}5\cdot 3=15 & & & -5(3)=-15 \\ 5(-3)=-15 & & & (-5)(-3)=15\end{array}\]

Notice that for multiplication of two signed numbers, when the

\[\begin{array}{l}\text{signs are the}\ \text{same}\text{, the product is}\ \text{positive}\text{.} \\ \text{signs are}\ \text{different}\text{, the product is}\ \text{negative}\text{.}\end{array}\]

What about division? Division is the inverse operation of multiplication. So, $15\div 3=5$ because $5\cdot 3=15.$ In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If you look at some examples of multiplying integers, you might figure out the rules for dividing integers.

\[\begin{array}{llllllll}5\cdot 3=15 & \text{so} & 15\div 3=5 & & & -5(3)=-15 & \text{so} & \ -15\div 3=-5 \\ (-5)(-3)=15 & \text{so} & 15\div (-3)=-5 & & & 5(-3)=-15 & \text{so} & -15\div (-3)=5\end{array}\]

Division follows the same rules as multiplication with regard to signs.

Example

Try it.

Multiply or divide: ⓐ $-100\div (-4)$ ⓑ $7\cdot 6$ ⓒ $4(-8)$ ⓓ $-27\div 3.$

Solution

ⓐ

	$-100\div (-4)$
Divide, with signs that are the same the quotient is positive.	$25$

ⓑ

	$7\cdot 6$
Multiply, with same signs.	$42$

ⓒ

	$4(-8)$
Multiply, with different signs.	$-32$

ⓓ

	$-27\div 3$
Divide, with different signs, the quotient is negative.	$-9$

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember Please Excuse My Dear Aunt Sally?

Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

Example

Try it.

Simplify: ⓐ ${(-2)}^{4}$ ⓑ $\text{-}{2}^{4}.$

Solution

Notice the difference in parts (a) and (b). In part (a), the exponent means to raise what is in the parentheses, the $-2$ to the 4^th power. In part (b), the exponent means to raise just the 2 to the 4^th power and then take the opposite.

ⓐ

	${(-2)}^{4}$
Write in expanded form.	$(-2)(-2)(-2)(-2)$
Multiply.	$4(-2)(-2)$
Multiply.	$-8(-2)$
Multiply.	$16$

ⓑ

	$-{2}^{4}$
Write in expanded form.	$-(2\cdot 2\cdot 2\cdot 2)$
We are asked to find the opposite of ${2}^{4}$.
Multiply.	$-(4\cdot 2\cdot 2)$
Multiply.	$-(8\cdot 2)$
Multiply.	$-16$

The last example showed us the difference between ${(-2)}^{4}$ and $\text{-}{2}^{4}.$ This distinction is important to prevent future errors. The next example reminds us to multiply and divide in order left to right.

Example

Try it.

Simplify: ⓐ $8(-9)\div {(-2)}^{3}$ ⓑ $-30\div 2+(-3)(-7).$

Solution

ⓐ

	$8(-9)\div {(-2)}^{3}$
Exponents first.	$8(-9)\div (-8)$
Multiply.	$-72\div (-8)$
Divide.	$9$

ⓑ

	$-30\div 2+(-3)(-7)$
Multiply and divide left to right, so divide first.	$-15+(-3)(-7)$
Multiply.	$-15+21$
Add.	$6$

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

Example

Try it.

Evaluate $4{x}^{2}-2xy+3{y}^{2}$ when $x=2,y=-1.$

Solution


	$\$
Simplify exponents.
Multiply.
Subtract.
Add.

Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

Example

Try it.

Translate and simplify: the sum of 8 and $-12,$ increased by $3.$

Solution

	$\text{the}\ \text{sum}\ \underset{\text{-}}{\text{of}}\ 8\ \underset{\text{---}}{\text{and}}\ -12\ \text{increased by 3}$
Translate.	$[8+(-12)]+3$
Simplify. Be careful not to confuse the brackets with an absolute value sign.	$(-4)+3$
Add.	$-1$

Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

How to Solve Application Problems Using Integers

Try it.

In the morning, the temperature in Kendallville, Indiana was 11 degrees. By mid-afternoon, the temperature had dropped to $-9$ degrees. What was the difference in the morning and afternoon temperatures?

Solution

Key Concepts

Opposite Notation
\[\begin{array}{l}\text{-}a\ \text{means the opposite of the number}\ a \\ \text{The notation}\ \text{-}a\ \text{is read as “the opposite of}\ a\text{.”}\end{array}\]
Absolute Value
The absolute value of a number is its distance from 0 on the number line.
The absolute value of a number n is written as $|n|$ and $|n|\ge 0$ for all numbers.
Absolute values are always greater than or equal to zero.
Grouping Symbols
\[\begin{array}{llllllllll}\text{Parentheses} & & & (\ ) & & & \text{Braces} & & & \{\ \} \\ \text{Brackets} & & & [\ ] & & & \text{Absolute value} & & & \ |\ |\end{array}\]
Subtraction Property
$\ a-b=a+(\text{-}b)$
Subtracting a number is the same as adding its opposite.
Multiplication and Division of Signed Numbers
For multiplication and division of two signed numbers:

Same signs Result
•  Two positives Positive
•  Two negatives Positive

$\$If the signs are the same, the result is positive.

Different signs Result
•  Positive and negative Negative
•  Negative and positive Negative

$\$If the signs are different, the result is negative.
Multiplication by $-1$
$\ -1a=\text{-}a$
Multiplying a number by $-1$ gives its opposite.
How to Use Integers in Applications.
1. Read the problem. Make sure all the words and ideas are understood
2. Identify what we are asked to find.
3. Write a phrase that gives the information to find it.
4. Translate the phrase to an expression.
5. Simplify the expression.
6. Answer the question with a complete sentence.

Integers

Simplify Expressions with Absolute Value

In the following exercises, fill in $<,>,$ or $=$ for each of the following pairs of numbers.

Try it.

ⓐ $|-7|___-|-7|$
ⓑ $6___-|-6|$
ⓒ $|-11|___-11$
ⓓ $\text{-}(-13)___-|-13|$

Solution

ⓐ $>$ ⓑ $>$ ⓒ $>$ ⓓ $>$

Try it.

ⓐ $\text{-}|-9|___|-9|$
ⓑ $-8___|-8|$
ⓒ $|-1|___-1$
ⓓ $\text{-}(-14)___-|-14|$

Try it.

ⓐ $\text{-}|2|___-|-2|$
ⓑ $-12___-|-12|$
ⓒ $|-3|___-3$
ⓓ $|-19|___-(-19)$

Solution

ⓐ $=$ ⓑ $=$ ⓒ $>$ ⓓ $=$

Try it.

ⓐ $\text{-}|-4|___-|4|$
ⓑ $5___-|-5|$
ⓒ $\text{-}|-10|___-10$
ⓓ $\text{-}|-0|___-(-0)$

In the following exercises, simplify.

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.

$18-|12-4(4-1)+3|$

Solution

Try it.

$27-|19+4(3-1)-7|$

Try it.

$10-3|9-3(3-1)|$

Solution

Try it.

$13-2|11-2(5-2)|$

Add and Subtract Integers

In the following exercises, simplify each expression.

Try it.

ⓐ $-7+(-4)$
ⓑ $-7+4$
ⓒ $7+(-4).$

Solution

ⓐ $-11$ ⓑ $-3$ ⓒ $3$

Try it.

ⓐ $-5+(-9)$
ⓑ $-5+9$
ⓒ $5+(-9)$

Try it.

$48+(-16)$

Solution

Try it.

$34+(-19)$

Try it.

$-14+(-12)+4$

Solution

$-22$

Try it.

$-17+(-18)+6$

Try it.

$19+2(-3+8)$

Solution

$29$

Try it.

$24+3(-5+9)$

Try it.

ⓐ $13-7$
ⓑ $-13-(-7)$
ⓒ $-13-7$
ⓓ $13-(-7)$

Solution

ⓐ 6 ⓑ $-6$ ⓒ $-20$ ⓓ $20$

Try it.

ⓐ $15-8$
ⓑ $-15-(-8)$
ⓒ $-15-8$
ⓓ $15-(-8)$

Try it.

$-17-42$

Solution

$-59$

Try it.

$-58-(-67)$

Try it.

$-14-(-27)+9$

Solution

Try it.

$64+(-17)-9$

Try it.

ⓐ $44-28$ ⓑ $44+(-28)$

Solution

ⓐ 16 ⓑ 16

Try it.

ⓐ $35-16$ ⓑ $35+(-16)$

Try it.

ⓐ $27-(-18)$ ⓑ $27+18$

Solution

ⓐ 45 ⓑ 45

Try it.

ⓐ $46-(-37)$ ⓑ $46+37$

Try it.

$(2-7)-(3-8)$

Solution

Try it.

$(1-8)-(2-9)$

Try it.

$\text{-}(6-8)-(2-4)$

Solution

Try it.

$\text{-}(4-5)-(7-8)$

Try it.

$25-[10-(3-12)]$

Solution

Try it.

$32-[5-(15-20)]$

Multiply and Divide Integers

In the following exercises, multiply or divide.

Try it.

ⓐ $-4\cdot 8$
ⓑ $13(-5)$
ⓒ $-24\div 6$
ⓓ $-52\div (-4)$

Solution

ⓐ $-32$ ⓑ $-65$ ⓒ $-4$
ⓓ $13$

Try it.

ⓐ $-3\cdot 9$
ⓑ $9(-7)$
ⓒ $35\div (-7)$
ⓓ $-84\div (-6)$

Try it.

ⓐ $-28\div 7$
ⓑ $-180\div 15$
ⓒ $3(-13)$
ⓓ $-1(-14)$

Solution

ⓐ $-4$ ⓑ $-12$ ⓒ $-39$
ⓓ $14$

Try it.

ⓐ $-36\div 4$
ⓑ $-192\div 12$
ⓒ $9(-7)$
ⓓ $-1(-19)$

Simplify and Evaluate Expressions with Integers

In the following exercises, simplify each expression.

Try it.

ⓐ ${(-2)}^{6}$ ⓑ $\text{-}{2}^{6}$

Solution

ⓐ $64$ ⓑ $-64$

Try it.

ⓐ ${(-3)}^{5}$ ⓑ $\text{-}{3}^{5}$

Try it.

$5(-6)+7(-2)-3$

Solution

$-47$

Try it.

$8(-4)+5(-4)-6$

Try it.

$-3(-5)(6)$

Solution

$90$

Try it.

$-4(-6)(3)$

Try it.

$(8-11)(9-12)$

Solution

$9$

Try it.

$(6-11)(8-13)$

Try it.

$26-3(2-7)$

Solution

$41$

Try it.

$23-2(4-6)$

Try it.

$65\div (-5)+(-28)\div (-7)$

Solution

$-9$

Try it.

$52\div (-4)+(-32)\div (-8)$

Try it.

$9-2[3-8(-2)]$

Solution

$-29$

Try it.

$11-3[7-4(-2)]$

Try it.

$8-|2-4(4-1)+3|$

Solution

$1$

Try it.

$7-|5-3(4-1)-6|$

Try it.

$9-3|2(2-6)-(3-7)|$

Solution

$-3$

Try it.

$5-2|2(1-4)-(2-5)|$

Try it.

${(-3)}^{2}-24\div (8-2)$

Solution

$5$

Try it.

${(-4)}^{2}-32\div (12-4)$

In the following exercises, evaluate each expression.

Try it.

$y+(-14)$ when
ⓐ $y=-33$ ⓑ $y=30$

Solution

ⓐ $-47$ ⓑ $16$

Try it.

$x+(-21)$ when
ⓐ $x=-27$ ⓑ $x=44$

Try it.

${(x+y)}^{2}$ when
$x=-3,y=14$

Solution

$121$

Try it.

${(y+z)}^{2}$ when
$y=-3,z=15$

Try it.

$9a-2b-8$ when
$a=-6$ and $b=-3$

Solution

$-56$

Try it.

$7m-4n-2$ when
$m=-4$ and $n=-9$

Try it.

$3{x}^{2}-4xy+2{y}^{2}$ when
$x=-2,y=-3$

Solution

$6$

Try it.

$4{x}^{2}-xy+3{y}^{2}$ when
$x=-3,y=-2$

Translate English Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

Try it.

the sum of 3 and $-15,$ increased by 7

Solution

$(3+(-15))+7;-5$

Try it.

the sum of $-8$ and $-9,$ increased by $23$

Try it.

ⓐ the difference of $10$ and $-18$
ⓑ subtract $11$ from $-25$

Solution

ⓐ $10-(-18);28$
ⓑ $-25-11;-36$

Try it.

ⓐ the difference of $-5$ and $-30$
ⓑ subtract $-6$ from $-13$

Try it.

the quotient of $-6$ and the sum of $a$ and $b$

Solution

$\frac{-6}{a+b}$

Try it.

the product of $-13$ and the difference of $c$ and $d$

Use Integers in Applications

In the following exercises, solve.

Try it.

Temperature On January 15, the high temperature in Anaheim, California, was 84°. That same day, the high temperature in Embarrass, Minnesota, was $\text{-}12\text{^{\circ}}.$ What was the difference between the temperature in Anaheim and the temperature in Embarrass?

Solution

$96\text{^{\circ}}$

Try it.

Temperature On January 21, the high temperature in Palm Springs, California, was $89\text{^{\circ}},$ and the high temperature in Whitefield, New Hampshire, was $\text{-}31\text{^{\circ}}.$ What was the difference between the temperature in Palm Springs and the temperature in Whitefield?

Try it.

Football On the first down, the Chargers had the ball on their 25-yard line. They lost 6 yards on the first-down play, gained 10 yards on the second-down play, and lost 8 yards on the third-down play. What was the yard line at the end of the third-down play?

Solution

21 yards

Try it.

Football On first down, the Steelers had the ball on their 30-yard line. They gained 9 yards on the first-down play, lost 14 yards on the second-down play, and lost 2 yards on the third-down play. What was the yard line at the end of the third-down play?

Try it.

Checking Account Mayra has $124 in her checking account. She writes a check for $152. What is the new balance in her checking account?

Solution

$\text{-}\text{\$}28$

Try it.

Checking Account Reymonte has a balance of $\text{-}\text{\$}49$ in his checking account. He deposits $281 to the account. What is the new balance?

Condensed — the full section is in OpenStax Intermediate Algebra 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.

Fill in $<,>,$ or $=$ for each of the following pairs of numbers:
ⓐ $|-5|__-|-5|$ ⓑ $8__-|-8|$ ⓒ $-9__-|-9|$ ⓓ $\text{-}(-16)__|-16|.$

答えを明らかにしろ

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

ⓑ
$8__-|-8|$
Simplify. $8__-8$
Order. $8>-8$
$8>-|-8|$

ⓒ
$-9__-|-9|$
Simplify. $-9__-9$
Order. $-9=-9$
$-9=-|-9|$

ⓓ
$-(-16)__|-16|$
Simplify. $\ 16__16$
Order. $\ 16=16$
$-(-16)=|-16|$
Fill in $<,>,$ or $=$ for each of the following pairs of numbers:
ⓐ $-9__-|-9|$ ⓑ $2__-|-2|$ ⓒ $-8__|-8|$ ⓓ $\text{-}(-9)__|-9|.$

答えを明らかにしろ

ⓐ $=$ ⓑ $>$ ⓒ $<$
ⓓ $=$
Fill in $<,>,$ or $=$ for each of the following pairs of numbers:
ⓐ $7__-|-7|$ ⓑ $\text{-}(-10)__|-10|$ ⓒ $|-4|__-|-4|$ ⓓ $-1__|-1|.$

答えを明らかにしろ

ⓐ $>$ ⓑ $=$ ⓒ $>$
ⓓ $<$

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

答えを明らかにしろ

	$24-\|19-3(6-2)\|$
Work inside parentheses first:
subtract 2 from 6.	$24-\|19-3(4)\|$
Multiply 3(4).	$24-\|19-12\|$
Subtract inside the absolute value bars.	$24-\|7\|$
Take the absolute value.	$24-7$
Subtract.	$17$

Simplify:$19-|11-4(3-1)|.$

答えを明らかにしろ

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

答えを明らかにしろ

9
Add: ⓐ $-1+(-4)$ ⓑ $-1+5$ ⓒ $1+(-5).$

答えを明らかにしろ

ⓐ

1 negative plus 4 negatives is 5 negatives

ⓑ

There are more positives, so the sum is positive.
ⓒ

There are more negatives, so the sum is negative.
Subtract: ⓐ $3-1$ ⓑ $-3-(-1)$ ⓒ $-3-1$ ⓓ $3-(-1).$

答えを明らかにしろ

ⓐ

$\$
Take 1 negative from 3 negatives and get 2 negatives positives.

ⓑ

Take 1 positive from 3 negatives and get 2 negatives.

ⓒ

$\$
Take 1 positive from the one added neutral pair.

ⓓ

$\$
Take 1 negative from the one added neutral pair.
Simplify: ⓐ $13-8$ and $13+(-8)$ ⓑ $-17-9$ and $-17+(-9)$ ⓒ $9-(-15)$ and $9+15$ ⓓ $-7-(-4)$ and $-7+4.$

答えを明らかにしろ

ⓐ

Subtract. $\begin{array}{lll}13-8 & \ \text{and}\ & 13+(-8) \\ 5 & & 5\end{array}$

ⓑ
Subtract. $\begin{array}{lll}-17-9 & \ \text{and}\ & -17+(-9) \\ -26 & & -26\end{array}$

ⓒ
Subtract. $\begin{array}{lll}9-(-15) & \ \text{and}\ & 9+15 \\ 24 & & 24\end{array}$

ⓓ

Subtract. $\begin{array}{lll}-7-(-4) & \ \text{and}\ & -7+4 \\ -3 & & -3\end{array}$
Simplify: ⓐ $21-13$ and $21+(-13)$ ⓑ $-11-7$ and $-11+(-7)$ ⓒ $6-(-13)$ and $6+13$ ⓓ $-5-(-1)$ and $-5+1.$

答えを明らかにしろ

ⓐ $8,8$ ⓑ $-18,$ $-18$
ⓒ $19,19$ ⓓ $-4,$ $-4$
Simplify: ⓐ $15-7$ and $15+(-7)$ ⓑ $-14-8$ and $-14+(-8)$ ⓒ $4-(-19)$ and $4+19$ ⓓ $-4-(-7)$ and $-4+7.$

答えを明らかにしろ

ⓐ $8,8$ ⓑ $-22,$ $-22$
ⓒ $23,23$ ⓓ $3,3$
Simplify: $7-(-4-3)-9.$

答えを明らかにしろ

$7-(-4-3)-9$
Simplify inside the parentheses first. $7-(-7)-9$
Subtract left to right. $14-9$
Subtract. $5$
Simplify: $8-(-3-1)-9.$

答えを明らかにしろ

3
Simplify: $12-(-9-6)-14.$

答えを明らかにしろ

13
Multiply or divide: ⓐ $-100\div (-4)$ ⓑ $7\cdot 6$ ⓒ $4(-8)$ ⓓ $-27\div 3.$

答えを明らかにしろ

ⓐ

$-100\div (-4)$
Divide, with signs that are the same the quotient is positive. $25$

ⓑ

$7\cdot 6$
Multiply, with same signs. $42$

ⓒ

$4(-8)$
Multiply, with different signs. $-32$

ⓓ

$-27\div 3$
Divide, with different signs, the quotient is negative. $-9$
Multiply or divide: ⓐ $-115\div (-5)$ ⓑ $5\cdot 12$ ⓒ $9(-7)$ ⓓ $-63\div 7.$

答えを明らかにしろ

ⓐ 23 ⓑ 60 ⓒ $-63$ ⓓ $-9$
Simplify: ⓐ ${(-2)}^{4}$ ⓑ $\text{-}{2}^{4}.$

答えを明らかにしろ

Notice the difference in parts (a) and (b). In part (a), the exponent means to raise what is in the parentheses, the $-2$ to the 4^th power. In part (b), the exponent means to raise just the 2 to the 4^th power and then take the opposite.
ⓐ

${(-2)}^{4}$
Write in expanded form. $(-2)(-2)(-2)(-2)$
Multiply. $4(-2)(-2)$
Multiply. $-8(-2)$
Multiply. $16$

ⓑ

$-{2}^{4}$
Write in expanded form. $-(2\cdot 2\cdot 2\cdot 2)$
We are asked to find the opposite of ${2}^{4}$.
Multiply. $-(4\cdot 2\cdot 2)$
Multiply. $-(8\cdot 2)$
Multiply. $-16$
Simplify: ⓐ ${(-3)}^{4}$ ⓑ $\text{-}{3}^{4}.$

答えを明らかにしろ

ⓐ 81 ⓑ $-81$
Simplify: ⓐ ${(-7)}^{2}$ ⓑ $\text{-}{7}^{2}.$

答えを明らかにしろ

ⓐ 49 ⓑ $-49$
Simplify: ⓐ $8(-9)\div {(-2)}^{3}$ ⓑ $-30\div 2+(-3)(-7).$

答えを明らかにしろ

ⓐ

$8(-9)\div {(-2)}^{3}$
Exponents first. $8(-9)\div (-8)$
Multiply. $-72\div (-8)$
Divide. $9$

ⓑ

$-30\div 2+(-3)(-7)$
Multiply and divide left to right, so divide first. $-15+(-3)(-7)$
Multiply. $-15+21$
Add. $6$
Simplify: ⓐ $12(-9)\div {(-3)}^{3}$ ⓑ $-27\div 3+(-5)(-6).$

答えを明らかにしろ

ⓐ 4 ⓑ 21
Simplify: ⓐ $18(-4)\div {(-2)}^{3}$ ⓑ $-32\div 4+(-2)(-7).$

答えを明らかにしろ

ⓐ 9 ⓑ 6
Evaluate $4{x}^{2}-2xy+3{y}^{2}$ when $x=2,y=-1.$

答えを明らかにしろ

$\$
Simplify exponents.
Multiply.
Subtract.
Add.
Evaluate: $3{x}^{2}-2xy+6{y}^{2}$ when $x=1,y=-2.$

答えを明らかにしろ

31
Evaluate: $4{x}^{2}-xy+5{y}^{2}$ when $x=-2,y=3.$

答えを明らかにしろ

67

Translate and simplify: the sum of 8 and $-12,$ increased by $3.$

答えを明らかにしろ

	$\text{the}\ \text{sum}\ \underset{\text{-}}{\text{of}}\ 8\ \underset{\text{---}}{\text{and}}\ -12\ \text{increased by 3}$
Translate.	$[8+(-12)]+3$
Simplify. Be careful not to confuse the brackets with an absolute value sign.	$(-4)+3$
Add.	$-1$

Translate and simplify the sum of 9 and $-16,$ increased by 4.

答えを明らかにしろ

$(9+(-16))+4;-3$
Translate and simplify the sum of $-8$ and $-12,$ increased by 7.

答えを明らかにしろ

$(-8+(-12))+7;-13$
In the morning, the temperature in Kendallville, Indiana was 11 degrees. By mid-afternoon, the temperature had dropped to $-9$ degrees. What was the difference in the morning and afternoon temperatures?
In the morning, the temperature in Anchorage, Alaska was $15$ degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

答えを明らかにしろ

The difference in temperatures was 45 degrees.
The temperature in Denver was $-6$ degrees at lunchtime. By sunset the temperature had dropped to $-15$ degrees. What was the difference in the lunchtime and sunset temperatures?

答えを明らかにしろ

The difference in temperatures was 9 degrees.
ⓐ $|-7|___-|-7|$
ⓑ $6___-|-6|$
ⓒ $|-11|___-11$
ⓓ $\text{-}(-13)___-|-13|$

答えを明らかにしろ

ⓐ $>$ ⓑ $>$ ⓒ $>$ ⓓ $>$
ⓐ $\text{-}|2|___-|-2|$
ⓑ $-12___-|-12|$
ⓒ $|-3|___-3$
ⓓ $|-19|___-(-19)$

答えを明らかにしろ

ⓐ $=$ ⓑ $=$ ⓒ $>$ ⓓ $=$

Symbols used here

\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.

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

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

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Integers

Simplify expressions with absolute value
Add and subtract integers
Multiply and divide integers
Simplify expressions with integers
Evaluate variable expressions with integers
Translate phrases to expressions with integers
Use integers in applications

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

あなた自身を試してみてください

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

ここに Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

	\(\|-5\|__-\|-5\|\)
Simplify.	\(\ 5__-5\)
Order.	\(\ 5>-5\)
	\(\|-5\|>-\|-5\|\)

	\(8__-\|-8\|\)
Simplify.	\(8__-8\)
Order.	\(8>-8\)
	\(8>-\|-8\|\)

	\(-9__-\|-9\|\)
Simplify.	\(-9__-9\)
Order.	\(-9=-9\)
	\(-9=-\|-9\|\)

	\(-(-16)__\|-16\|\)
Simplify.	\(\ 16__16\)
Order.	\(\ 16=16\)
	\(-(-16)=\|-16\|\)

	\({(-2)}^{4}\)
Write in expanded form.	\((-2)(-2)(-2)(-2)\)
Multiply.	\(4(-2)(-2)\)
Multiply.	\(-8(-2)\)
Multiply.	\(16\)

	\(\\)
Take 1 negative from 3 negatives and get 2 negatives positives.

Same signs	Result
• Two positives	Positive
• Two negatives	Positive

Different signs	Result
• Positive and negative	Negative
• Negative and positive	Negative

	\(-100\div (-4)\)
Divide, with signs that are the same the quotient is positive.	\(25\)

	\(-27\div 3\)
Divide, with different signs, the quotient is negative.	\(-9\)

	\(-{2}^{4}\)
Write in expanded form.	\(-(2\cdot 2\cdot 2\cdot 2)\)
We are asked to find the opposite of \({2}^{4}\).
Multiply.	\(-(4\cdot 2\cdot 2)\)
Multiply.	\(-(8\cdot 2)\)
Multiply.	\(-16\)

	\(8(-9)\div {(-2)}^{3}\)
Exponents first.	\(8(-9)\div (-8)\)
Multiply.	\(-72\div (-8)\)
Divide.	\(9\)

	\(-30\div 2+(-3)(-7)\)
Multiply and divide left to right, so divide first.	\(-15+(-3)(-7)\)
Multiply.	\(-15+21\)
Add.	\(6\)

	\(\text{the}\ \text{sum}\ \underset{\text{-}}{\text{of}}\ 8\ \underset{\text{---}}{\text{and}}\ -12\ \text{increased by 3}\)
Translate.	\([8+(-12)]+3\)
Simplify. Be careful not to confuse the brackets with an absolute value sign.	\((-4)+3\)
Add.	\(-1\)

Same signs	Result
• Two positives	Positive
• Two negatives	Positive

Different signs	Result
• Positive and negative	Negative
• Negative and positive	Negative

	\(24-\|19-3(6-2)\|\)
Work inside parentheses first:
subtract 2 from 6.	\(24-\|19-3(4)\|\)
Multiply 3(4).	\(24-\|19-12\|\)
Subtract inside the absolute value bars.	\(24-\|7\|\)
Take the absolute value.	\(24-7\)
Subtract.	\(17\)

	\(7-(-4-3)-9\)
Simplify inside the parentheses first.	\(7-(-7)-9\)
Subtract left to right.	\(14-9\)
Subtract.	\(5\)