Add and Subtract Integers

Use Negatives and Opposites

Our work so far has only included the counting numbers and the whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. Negative numbers are numbers less than \(0.\) The negative numbers are to the left of zero on the number line.

The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number.

Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative.

Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value.

Remember that we use the notation:

a < b (read “a is less than b”) when a is to the left of b on the number line.

a > b (read “a is greater than b”) when a is to the right of b on the number line.

Example

Try it.

Order each of the following pairs of numbers, using < or >: ⓐ \(14\underline{\hspace{1.5em}}6\) ⓑ \(-1\underline{\hspace{1.5em}}9\) ⓒ \(-1\underline{\hspace{1.5em}}-4\) ⓓ \(2\underline{\hspace{1.5em}}-20.\)

Solution

It may be helpful to refer to the number line shown.

ⓐ 14 is to the right of 6 on the number line.	\(\begin{array}{l}14\underline{\hspace{1.5em}}6 \\ 14>6\end{array}\)
ⓑ −1 is to the left of 9 on the number line.	\(\begin{array}{l}-1\underline{\hspace{1.5em}}9 \\ -1<9\end{array}\)
ⓒ −1 is to the right of −4 on the number line.	\(\begin{array}{l}-1\underline{\hspace{1.5em}}-4 \\ -1>-4\end{array}\)
ⓓ 2 is to the right of −20 on the number line.	\(\begin{array}{l}2\underline{\hspace{1.5em}}-20 \\ 2>-20\end{array}\)

\(10-4\)	Between two numbers, it indicates the operation of subtraction. We read \(10-4\) as "10 minus 4."
\(-8\)	In front of a number, it indicates a negative number. We read −8 as "negative eight."
\(\text{-}x\)	In front of a variable, it indicates the opposite. We read \(\text{-}x\) as "the opposite of \(x\)."
\(\text{-}(-2)\)	Here there are two "−" signs. The one in the parentheses tells us the number is negative 2. The one outside the parentheses tells us to take the opposite of −2. We read \(\text{-}(-2)\) as "the opposite of negative two."

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Simplify: Expressions with Absolute Value

We saw that numbers such as \(2\ \text{and}\ -2\) are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

For example,

• \(-5\ \text{is}\ 5\) units away from \(0,\) so \(|-5|=5.\)
• \(\ 5\ \text{is}\ 5\) units away from \(0,\) so \(|5|=5.\)

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 \(0\ \text{to}\ 0\) on the number line is zero units.

Mathematicians say it more precisely, “absolute values are always non-negative.” Non-negative means greater than or equal to zero.

Example

Try it.

Simplify: ⓐ \(|3|\) ⓑ \(|-44|\) ⓒ \(|0|\).

Solution

The absolute value of a number is the distance between the number and zero. Distance is never negative, so the absolute value is never negative.

ⓐ \(|3|\)
\(\ 3\)

ⓑ \(|-44|\)
\(\ 44\)

ⓒ \(|0|\)
\(\ 0\)

In the next example, we’ll order expressions with absolute values. Remember, positive numbers are always greater than negative numbers!

Example

Try it.

Fill in \(<,>,\ \text{or}\ =\) for each of the following pairs of numbers:

ⓐ \(|-5|\underline{\hspace{1.5em}}-|-5|\) ⓑ \(8\underline{\hspace{1.5em}}-|-8|\) ⓒ \(-9\underline{\hspace{1.5em}}-|-9|\) ⓓ \(\text{-}(-16)\underline{\hspace{1.5em}}-|-16|\)

Solution

ⓐ Simplify. Order.	\(\begin{array}{lll}|-5| & \underline{\hspace{1.5em}} & -|-5| \\ 5 & \underline{\hspace{1.5em}} & -5 \\ 5 & > & -5 \\ |-5| & > & -|-5|\end{array}\)
ⓑ Simplify. Order.	\(\begin{array}{lll}8 & \underline{\hspace{1.5em}} & -|-8| \\ 8 & \underline{\hspace{1.5em}} & -8 \\ 8 & > & -8 \\ 8 & > & -|-8|\end{array}\)
ⓒ Simplify. Order.	\(\begin{array}{lll}-9 & \underline{\hspace{1.5em}} & -|-9| \\ -9 & \underline{\hspace{1.5em}} & -9 \\ -9 & = & -9 \\ -9 & = & -|-9|\end{array}\)
ⓓ Simplify. Order.	\(\begin{array}{lll}\text{-}(-16) & \underline{\hspace{1.5em}} & -|-16| \\ 16 & \underline{\hspace{1.5em}} & -16 \\ 16 & > & -16 \\ \text{-}(-16) & > & -|-16|\end{array}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Add Integers

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 the four addition facts using the numbers \(5,-5\) and \(3,-3.\)

\[\begin{array}{llllllllll}5+3 & & & -5+(-3) & & & -5+3 & & & 5+(-3)\end{array}\]

To add \(5+3,\) we realize that \(5+3\) means the sum of 5 and 3.

We start with 5 positives.
And then we add 3 positives.
We now have 8 positives. The sum of 5 and 3 is 8.

Now we will add \(-5+(-3).\) Watch for similarities to the last example \(5+3=8.\)

To add \(-5+(-3),\) we realize this means the sum of \(-5\ \text{and}\ -3.\)

We start with 5 negatives.
And then we add 3 negatives.
We now have 8 negatives. The sum of −5 and −3 is −8.

• The first example adds 5 positives and 3 positives, both positives.
• The second example adds 5 negatives and 3 negatives, both negatives.

Example

Try it.

Add: ⓐ \(1+4\) ⓑ \(-1+(-4).\)

Solution

ⓐ

1 positive plus 4 positives is 5 positives.

ⓑ

1 negative plus 4 negatives is 5 negatives.

	−5 + 3 means the sum of −5 and 3.
We start with 5 negatives.
And then we add 3 positives.
We remove any neutral pairs.
We have 2 negatives left.
The sum of −5 and 3 is −2.	−5 + 3 = −2

	5 + (−3) means the sum of 5 and −3.
We start with 5 positives.
And then we add 3 negatives.
We remove any neutral pairs.
We have 2 positives left.
The sum of 5 and −3 is 2.	5 + (−3) = 2

Example

Try it.

Add: ⓐ \(-1+5\) ⓑ \(1+(-5).\)

Solution

ⓐ

	−1 + 5
There are more positives, so the sum is positive.	4

ⓑ

	1 + (−5)
There are more negatives, so the sum is negative.	−4

\[37+(\text{-}53)=-16\]\[-74+(\text{-}27)=-101\]

Example

Try it.

Simplify: \(-5+3(-2+7).\)

Solution

	\(-5+3(-2+7)\)
Simplify inside the parentheses.	\(-5+3(5)\)
Multiply.	\(-5+15\)
Add left to right.	\(10\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Subtract Integers

We will continue to use counters to model the subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.

Perhaps when you were younger, you read \(\text{“}5-3\text{”}\) as \(\text{“}5\) take away \(3.\text{”}\) When you use counters, you can think of subtraction the same way!

We will model the four subtraction facts using the numbers \(5\) and \(3.\)

\[\begin{array}{llllllllll}5-3 & & & -5-(-3) & & & -5-3 & & & 5-(-3)\end{array}\]

To subtract \(5-3,\) we restate the problem as \(\text{“}5\) take away \(3.\text{”}\)

We start with 5 positives.
We ‘take away’ 3 positives.
We have 2 positives left.
The difference of 5 and 3 is 2.	2

Now we will subtract \(-5-(-3).\) Watch for similarities to the last example \(5-3=2.\)

To subtract \(-5-(-3),\) we restate this as \(\text{“}-5\) take away \(-3\text{”}\)

We start with 5 negatives.
We ‘take away’ 3 negatives.
We have 2 negatives left.
The difference of −5 and −3 is −2.	−2

Notice that these two examples are much alike: The first example, we subtract 3 positives from 5 positives and end up with 2 positives.

Example

Try it.

Subtract: ⓐ \(7-5\) ⓑ \(-7-(-5).\)

Solution

ⓐ Take 5 positive from 7 positives and get 2 positives.	\(\begin{array}{l}7-5 \\ 2\end{array}\)
ⓑ Take 5 negatives from 7 negatives and get 2 negatives.	\(\begin{array}{l}-7-(-5) \\ -2\end{array}\)

• To subtract \(-5-3,\) we restate it as \(-5\) take away 3.

	−5 − 3 means −5 take away 3.
We start with 5 negatives.
We now add the neutrals needed to get 3 positives.
We remove the 3 positives.
We are left with 8 negatives.
The difference of −5 and 3 is −8.	−5 − 3 = −8

	5 − (−3) means 5 take away −3.
We start with 5 positives.
We now add the needed neutrals pairs.
We remove the 3 negatives.
We are left with 8 positives.
The difference of 5 and −3 is 8.	5 − (−3) = 8

Example

Try it.

Subtract: ⓐ \(-3-1\) ⓑ \(3-(-1).\)

Solution

ⓐ

Take 1 positive from the one added neutral pair.

−3 − 1 −4

ⓑ

Take 1 negative from the one added neutral pair.

3 − (−1) 4

\[6-4\ \text{gives the same answer as}\ 6+(-4).\]

Example

Try it.

Simplify: ⓐ \(13-8\) and \(13+(-8)\) ⓑ \(-17-9\) and \(-17+(-9).\)

Solution

ⓐ Subtract.	\(\begin{array}{l}13-8 \\ 5\end{array}\)	\(\begin{array}{l}13+(-8) \\ 5\end{array}\)
ⓑ Subtract.	\(\begin{array}{l}-17-9 \\ -26\end{array}\)	\(\begin{array}{l}-17+(-9) \\ -26\end{array}\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

• Addition of Positive and Negative Integers
  \(\begin{array}{llll}5+3 & & & \ -5+(-3) \\ 8 & & & \ -8 \\ \text{both positive,} & & & \ \text{both negative,} \\ \text{sum positive} & & & \ \text{sum negative} \\ \\ \\ -5+3 & & & \ 5+(-3) \\ -2 & & & \ 2 \\ \text{different signs,} & & & \ \text{different signs,} \\ \text{more negatives} & & & \ \text{more positives} \\ \text{sum negative} & & & \ \text{sum positive}\end{array}\)
• Property of Absolute Value: \(|n|\ge 0\) for all numbers. Absolute values are always greater than or equal to zero!
• Subtraction of Integers
  \(\begin{array}{llll}5-3 & & & -5-(-3) \\ 2 & & & \ -2 \\ 5\ \text{positives} & & & 5\ \text{negatives} \\ \text{take away}\ 3\ \text{positives} & & & \text{take away}\ 3\ \text{negatives} \\ \text{2 positives} & & & \text{2 negatives} \\ \\ \\ -5-3 & & & 5-(-3) \\ -8 & & & \ 8 \\ 5\ \text{negatives, want to} & & & 5\ \text{positives, want to} \\ \text{subtract}\ 3\ \text{positives} & & & \text{subtract}\ 3\ \text{negatives} \\ \text{need neutral pairs} & & & \text{need neutral pairs}\end{array}\)
• Subtraction Property: Subtracting a number is the same as adding its opposite.

Add and Subtract Integers

Use Negatives and Opposites of Integers

In the following exercises, order each of the following pairs of numbers, using < or >.

Try it.

ⓐ \(9\underline{\hspace{1.5em}}4\)
ⓑ \(-3\underline{\hspace{1.5em}}6\)
ⓒ \(-8\underline{\hspace{1.5em}}-2\)
ⓓ \(1\underline{\hspace{1.5em}}-10\)

Solution

ⓐ > ⓑ < ⓒ < ⓓ >

Try it.

ⓐ \(-7\underline{\hspace{1.5em}}3\)
ⓑ \(-10\underline{\hspace{1.5em}}-5\)
ⓒ \(2\underline{\hspace{1.5em}}-6\)
ⓓ \(8\underline{\hspace{1.5em}}9\)

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

Try it.

ⓐ 2 ⓑ \(-6\)

Solution

ⓐ \(-2\) ⓑ 6

In the following exercises, simplify.

Try it.

\(\text{-}(-4)\)

Solution

4

Try it.

\(\text{-}(-8)\)

Try it.

\(\text{-}(-15)\)

Solution

15

Try it.

\(\text{-}(-11)\)

In the following exercises, evaluate.

Try it.

\(\text{-}c\) whenⓐ \(c=12\)ⓑ \(c=-12\)

Solution

ⓐ \(-12\) ⓑ 12

Try it.

\(\text{-}d\) when
ⓐ \(d=21\)
ⓑ \(d=-21\)

Simplify Expressions with Absolute Value

In the following exercises, simplify.

Try it.

ⓐ \(|-32|\)ⓑ \(|0|\)ⓒ \(|16|\)

Solution

ⓐ 32 ⓑ 0 ⓒ 16

Try it.

ⓐ \(|0|\)
ⓑ \(|-40|\)ⓒ \(|22|\)

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

Try it.

ⓐ \(-6\underline{\hspace{1.5em}}|-6|\)ⓑ \(\text{-}|-3|\underline{\hspace{1.5em}}-3\)

Solution

ⓐ < ⓑ \(=\)

Try it.

ⓐ \(|-5|\underline{\hspace{1.5em}}\text{-}|-5|\)ⓑ \(9\underline{\hspace{1.5em}}\text{-}|-9|\)

In the following exercises, simplify.

Try it.

\(\text{-}(-5)\ \text{and}\ \text{-}|-5|\)

Solution

\(5,-5\)

Try it.

\(\text{-}|-9|\ \text{and}\ \text{-}(-9)\)

Try it.

\(|15-7|-|14-6|\)

Solution

0

Try it.

\(|17-8|-|13-4|\)

Try it.

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

Solution

8

Try it.

\(18-|3(8-5)|\)

In the following exercises, evaluate.

Try it.

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

Solution

ⓐ \(-19\) ⓑ \(-33\)

Try it.

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

Add Integers

In the following exercises, simplify each expression.

Try it.

\(-21+(-59)\)

Solution

\(-80\)

Try it.

\(-35+(-47)\)

Try it.

\(48+(-16)\)

Solution

32

Try it.

\(34+(-19)\)

Try it.

\(-14+(-12)+4\)

Solution

\(-22\)

Try it.

\(-17+(-18)+6\)

Try it.

\(135+(-110)+83\)

Solution

108

Try it.

\(-38+27+(-8)+12\)

Try it.

\(19+2(-3+8)\)

Solution

29

Try it.

\(24+3(-5+9)\)

Subtract Integers

In the following exercises, simplify.

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.

ⓐ \(46-(-37)\)ⓑ \(46+37\)

In the following exercises, simplify each expression.

Try it.

\(15-(-12)\)

Solution

27

Try it.

\(14-(-11)\)

Try it.

\(-103-(-52)\)

Solution

\(-51\)

Try it.

\(-105-(-68)\)

Try it.

\(-45-(-54)\)

Solution

9

Try it.

\(-58-(-67)\)

Try it.

\(-14-(-27)+9\)

Solution

22

Try it.

\(64+(-17)-9\)

Try it.

\((2-7)-(3-8)\)

Solution

0

Try it.

\((1-8)-(2-9)\)

Try it.

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

Solution

4

Try it.

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

Try it.

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

Solution

6

Try it.

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

Try it.

\(6⋅3-4⋅3-7⋅2\)

Solution

\(-8\)

Try it.

\(5⋅7-8⋅2-4⋅9\)

Try it.

\({5}^{2}-{6}^{2}\)

Solution

\(-11\)

Try it.

\({6}^{2}-{7}^{2}\)

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

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Condensed: the full section is in OpenStax Elementary Algebra 2e.

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1. Order each of the following pairs of numbers, using < or >: ⓐ \(14\underline{\hspace{1.5em}}6\) ⓑ \(-1\underline{\hspace{1.5em}}9\) ⓒ \(-1\underline{\hspace{1.5em}}-4\) ⓓ \(2\underline{\hspace{1.5em}}-20.\)

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   It may be helpful to refer to the number line shown.

   ⓐ 14 is to the right of 6 on the number line.	\(\begin{array}{l}14\underline{\hspace{1.5em}}6 \\ 14>6\end{array}\)
   ⓑ −1 is to the left of 9 on the number line.	\(\begin{array}{l}-1\underline{\hspace{1.5em}}9 \\ -1<9\end{array}\)
   ⓒ −1 is to the right of −4 on the number line.	\(\begin{array}{l}-1\underline{\hspace{1.5em}}-4 \\ -1>-4\end{array}\)
   ⓓ 2 is to the right of −20 on the number line.	\(\begin{array}{l}2\underline{\hspace{1.5em}}-20 \\ 2>-20\end{array}\)

2. Order each of the following pairs of numbers, using \(<\) or \(>\text{:}\) ⓐ \(15\underline{\hspace{1.5em}}7\) ⓑ \(-2\underline{\hspace{1.5em}}5\) ⓒ \(-3\underline{\hspace{1.5em}}-7\)
   ⓓ \(5\underline{\hspace{1.5em}}-17.\)

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   ⓐ > ⓑ < ⓒ > ⓓ >

3. Order each of the following pairs of numbers, using \(<\) or \(>\text{:}\) ⓐ \(8\underline{\hspace{1.5em}}13\) ⓑ \(3\underline{\hspace{1.5em}}-4\) ⓒ \(-5\underline{\hspace{1.5em}}-2\)
   ⓓ \(9\underline{\hspace{1.5em}}-21.\)

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   ⓐ < ⓑ > ⓒ < ⓓ >

4. Find: ⓐ the opposite of 7 ⓑ the opposite of \(-10\) ⓒ \(\text{-}(-6).\)

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   ⓐ −7 is the same distance from 0 as 7, but on the opposite side of 0.	The opposite of 7 is −7.
   ⓑ 10 is the same distance from 0 as −10, but on the opposite side of 0.	The opposite of −10 is 10.
   ⓒ −(−6) (the opposite of –6).	The opposite of −(−6) is 6.

5. Find: ⓐ the opposite of 4 ⓑ the opposite of \(-3\) ⓒ \(\text{-}(-1).\)

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   ⓐ \(-4\) ⓑ 3 ⓒ 1

6. Find: ⓐ the opposite of 8 ⓑ the opposite of \(-5\) ⓒ \(\text{-}(-5).\)

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   ⓐ \(-8\) ⓑ 5 ⓒ 5

7. Evaluate ⓐ \(\text{-}x,\) when \(x=8\) ⓑ \(\text{-}x,\) when \(x=-8.\)

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   1. ⓐ
      

      	−x
      Write the opposite of 8.

   2. ⓑ
      

      	−x
      Write the opposite of −8.	8

8. Evaluate \(\text{-}n,\) when ⓐ \(n=4\) ⓑ \(n=-4.\)

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   ⓐ \(-4\) ⓑ 4

9. Evaluate \(\text{-}m,\) when ⓐ \(m=11\) ⓑ \(m=-11.\)

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   ⓐ \(-11\) ⓑ 11

10. Simplify: ⓐ \(|3|\) ⓑ \(|-44|\) ⓒ \(|0|\).

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    The absolute value of a number is the distance between the number and zero. Distance is never negative, so the absolute value is never negative.

    ⓐ \(|3|\)
    \(\ 3\)
    
    ⓑ \(|-44|\)
    \(\ 44\)
    
    ⓒ \(|0|\)
    \(\ 0\)

11. Simplify: ⓐ \(|4|\) ⓑ \(|-28|\) ⓒ \(|0|.\)

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    ⓐ 4 ⓑ 28 ⓒ 0

12. Simplify: ⓐ \(|-13|\) ⓑ \(|47|\) ⓒ \(|0|.\)

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    ⓐ 13 ⓑ 47 ⓒ 0

13. Fill in \(<,>,\ \text{or}\ =\) for each of the following pairs of numbers:

    ⓐ \(|-5|\underline{\hspace{1.5em}}-|-5|\) ⓑ \(8\underline{\hspace{1.5em}}-|-8|\) ⓒ \(-9\underline{\hspace{1.5em}}-|-9|\) ⓓ \(\text{-}(-16)\underline{\hspace{1.5em}}-|-16|\)

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    ⓐ Simplify. Order.	\(\begin{array}{lll}|-5| & \underline{\hspace{1.5em}} & -|-5| \\ 5 & \underline{\hspace{1.5em}} & -5 \\ 5 & > & -5 \\ |-5| & > & -|-5|\end{array}\)
    ⓑ Simplify. Order.	\(\begin{array}{lll}8 & \underline{\hspace{1.5em}} & -|-8| \\ 8 & \underline{\hspace{1.5em}} & -8 \\ 8 & > & -8 \\ 8 & > & -|-8|\end{array}\)
    ⓒ Simplify. Order.	\(\begin{array}{lll}-9 & \underline{\hspace{1.5em}} & -|-9| \\ -9 & \underline{\hspace{1.5em}} & -9 \\ -9 & = & -9 \\ -9 & = & -|-9|\end{array}\)
    ⓓ Simplify. Order.	\(\begin{array}{lll}\text{-}(-16) & \underline{\hspace{1.5em}} & -|-16| \\ 16 & \underline{\hspace{1.5em}} & -16 \\ 16 & > & -16 \\ \text{-}(-16) & > & -|-16|\end{array}\)

14. Fill in <, >, or \(=\) for each of the following pairs of numbers: ⓐ \(|-9|\underline{\hspace{1.5em}}-|-9|\) ⓑ \(2\underline{\hspace{1.5em}}-|-2|\) ⓒ \(-8\underline{\hspace{1.5em}}|-8|\)
    ⓓ \(\text{-}(-9)\underline{\hspace{1.5em}}-|-9|.\)

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    ⓐ > ⓑ > ⓒ < ⓓ >

15. Fill in <, >, or \(=\) for each of the following pairs of numbers: ⓐ \(7\underline{\hspace{1.5em}}-|-7|\) ⓑ \(\text{-}(-10)\underline{\hspace{1.5em}}-|-10|\)
    ⓒ \(|-4|\underline{\hspace{1.5em}}-|-4|\) ⓓ \(-1\underline{\hspace{1.5em}}|-1|.\)

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    ⓐ > ⓑ > ⓒ > ⓓ <

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

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    	\(24-|19-3(6-2)|\)
    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.	\(17\)

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

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    16

18. Simplify: \(9-|8-4(7-5)|.\)

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    9

19. Evaluate: ⓐ \(|x|\ \text{when}\ x=-35\) ⓑ \(|\text{-}y|\ \text{when}\ y=-20\) ⓒ \(\text{-}|u|\ \text{when}\ u=12\) ⓓ \(\text{-}|p|\ \text{when}\ p=-14.\)

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    ⓐ \(|x|\ \text{when}\ x=-35\)
    

    	\(|x|\)
    Take the absolute value.	35

    
    
    ⓑ \(|\text{-}y|\ \text{when}\ y=-20\)
    

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

    
    
    ⓒ \(\text{-}|u|\ \text{when}\ u=12\)
    

    	\(-|u|\)
    Take the absolute value.	\(-12\)

    
    
    ⓓ \(\text{-}|p|\ \text{when}\ p=-14\)
    

    	\(-|p|\)
    Take the absolute value.	\(-14\)

20. Evaluate: ⓐ \(|x|\ \text{when}\ x=-17\) ⓑ \(|\text{-}y|\ \text{when}\ y=-39\) ⓒ \(\text{-}|m|\ \text{when}\ m=22\) ⓓ \(\text{-}|p|\ \text{when}\ p=-11.\)

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    ⓐ \(17\) ⓑ \(39\) ⓒ \(-22\) ⓓ \(-11\)

21. Evaluate: ⓐ \(|y|\ \text{when}\ y=-23\) ⓑ \(|\text{-}y|\ \text{when}\ y=-21\) ⓒ \(\text{-}|n|\ \text{when}\ n=37\) ⓓ \(\text{-}|q|\ \text{when}\ q=-49.\)

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    ⓐ \(23\) ⓑ \(21\) ⓒ \(-37\) ⓓ \(-49\)

22. Add: ⓐ \(1+4\) ⓑ \(-1+(-4).\)

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    ⓐ
    

    1 positive plus 4 positives is 5 positives.

    ⓑ
    

    1 negative plus 4 negatives is 5 negatives.

23. Add: ⓐ \(2+4\) ⓑ \(-2+(-4).\)

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    ⓐ 6 ⓑ \(-6\)

24. Add: ⓐ \(2+5\) ⓑ \(-2+(-5).\)

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    ⓐ 7 ⓑ \(-7\)

25. Add: ⓐ \(-1+5\) ⓑ \(1+(-5).\)

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    ⓐ
    

    	−1 + 5
    There are more positives, so the sum is positive.	4

    
    
    ⓑ
    

    	1 + (−5)
    There are more negatives, so the sum is negative.	−4

26. Add: ⓐ \(-2+4\) ⓑ \(2+(-4).\)

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    ⓐ 2 ⓑ \(-2\)

27. Add: ⓐ \(-2+5\) ⓑ \(2+(-5).\)

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    ⓐ 3 ⓑ \(-3\)

28. Simplify: ⓐ \(19+(-47)\) ⓑ \(-14+(-36).\)

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    1. ⓐ Since the signs are different, we subtract \(\text{19 from 47}\text{.}\) The answer will be negative because there are more negatives than positives.
       \(\begin{array}{llll} & & & \ 19+(-47) \\ \text{Add.} & & & \ -28\end{array}\)
    2. ⓑ Since the signs are the same, we add. The answer will be negative because there are only negatives.
       \(\begin{array}{llll} & & & -14+(-36) \\ \text{Add.} & & & -50\end{array}\)

29. Simplify: ⓐ \(-31+(-19)\) ⓑ \(15+(-32).\)

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    ⓐ \(-50\) ⓑ \(-17\)

30. Simplify: ⓐ \(-42+(-28)\) ⓑ \(25+(-61).\)

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    ⓐ \(-70\) ⓑ \(-36\)

31. Simplify: \(-5+3(-2+7).\)

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    	\(-5+3(-2+7)\)
    Simplify inside the parentheses.	\(-5+3(5)\)
    Multiply.	\(-5+15\)
    Add left to right.	\(10\)

32. Simplify: \(-2+5(-4+7).\)

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    13

33. Simplify: \(-4+2(-3+5).\)

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    0

34. Subtract: ⓐ \(7-5\) ⓑ \(-7-(-5).\)

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    ⓐ Take 5 positive from 7 positives and get 2 positives.	\(\begin{array}{l}7-5 \\ 2\end{array}\)
    ⓑ Take 5 negatives from 7 negatives and get 2 negatives.	\(\begin{array}{l}-7-(-5) \\ -2\end{array}\)

35. Subtract: ⓐ \(6-4\) ⓑ \(-6-(-4).\)

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    ⓐ 2 ⓑ \(-2\)

36. Subtract: ⓐ \(7-4\) ⓑ \(-7-(-4).\)

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    ⓐ 3 ⓑ \(-3\)

37. Subtract: ⓐ \(-3-1\) ⓑ \(3-(-1).\)

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    ⓐ

    Take 1 positive from the one added neutral pair.

    −3 − 1 −4

    ⓑ

    Take 1 negative from the one added neutral pair.

    3 − (−1) 4

38. Subtract: ⓐ \(-6-4\) ⓑ \(6-(-4).\)

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    ⓐ \(-10\) ⓑ 10

39. Subtract: ⓐ \(-7-4\) ⓑ \(7-(-4).\)

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    ⓐ \(-11\) ⓑ 11

40. Simplify: ⓐ \(13-8\) and \(13+(-8)\) ⓑ \(-17-9\) and \(-17+(-9).\)

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    ⓐ Subtract.	\(\begin{array}{l}13-8 \\ 5\end{array}\)	\(\begin{array}{l}13+(-8) \\ 5\end{array}\)
    ⓑ Subtract.	\(\begin{array}{l}-17-9 \\ -26\end{array}\)	\(\begin{array}{l}-17+(-9) \\ -26\end{array}\)

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Используемые здесь символы

Нарисуйте любой символ для полного определения, картинку, и что каждая буква в ней означает.

\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\neqnot equal
The two sides are different.\sqrt{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.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

Как: Add and Subtract Integers

1. Use negatives and opposites
2. Simplify: expressions with absolute value
3. Add integers
4. Subtract integers
5. The first example adds 5 positives and 3 positives—both positives.
6. The second example adds 5 negatives and 3 negatives—both negatives.
7. To subtract

Вопросы, которые люди задают

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.

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