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Chapter 1
Introduction to
Algebraic
Expressions
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1.5 Addition of Real Numbers
• Adding with the Number Line
• Adding Without the Number Line
• Problem Solving
• Combining Like Terms
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Adding with a Number Line
To add a + b on a number line, we start at a and
move according to b.
a) If b is positive, we move to the right
(the positive direction).
b) If b is negative, we move to the left (the
negative direction).
c) If b is 0, we stay at a.
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Example
Add: −3 + 7.
Solution
Locate the first number −3, and then move 7 units
Start at −3
to the right
Move 7 units to the right.
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
−3 + 7 = 4
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Example
Add: −2 + (−3).
Solution
After locating −2, we move 3 units to the left.
Move
Start at −2
3 units to the left.
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
−2 + (−3) = −5
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1-5
Rules for Addition of Real Numbers
1. Positive numbers: Add as usual. The answer is positive.
2. Negative numbers: Add the absolute values and make the
answer negative.
3. A positive and a negative number: Subtract the smaller
absolute value from the greater absolute value.
a) If the positive number has the greater absolute value,
the answer is positive.
b) If the negative number has the greater absolute value,
the answer is negative.
c) If the numbers have the same absolute value, the
answer is 0.
4. One number is zero: The sum is the other number.
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Example
Add without using the number line.
Two negatives, add the
a) −9 + (−11)
absolute value, answer is −20.
b) −34 + 15
A negative and a positive,
subtract and the answer is
negative, −19.
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Example
Add without using the number line.
A negative and a positive,
c) −2.3 + 7.4
subtract and the answer is
positive, 5.1.
d) 2.4 + (−2.4)
A negative and a positive,
subtract and the answer is 0.
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Example
Add: 17 + (−3) + 9 + 16 + (−4) + (−12).
Solution
17 + (−3) + 9 + 16 + (−4) + (−12)
= 17 + 9 + 16 + (−3)+ (−4) + (−12) Using the commutative law
= (17 + 9 + 16) +[(−3)+ (−4) + (−12)] Using the associative law
= 42 + (−19)
= 23
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Example
During the first two weeks of the semester, 6 students
withdrew from Mr. Lange’s algebra class, 9 students were
added to the class, and 4 students were dropped as “noshows.” By how many students did the original class size
change?
The 1st plus the 2nd plus the 3rd is the total
change
6
 change
 change = change


9
( 4)
 Total change
The original class size dropped by one.
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Combining Like Terms
• When two terms have variable factors that are
exactly the same, the terms are called like or
similar terms.
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Example
Combine like terms −5x + 7x.
Solution −5x + 7x = (−5 + 7)x
= 2x
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Example
Combine like terms: 3a + (−4b) + (−8a) + 10b
Solution
3a + (−4b) + (−8a) + 10b
= 3a + (−8a) + (−4b) + 10b
= (3 +(−8))a + (−4 + 10)b
= −5a + 6b
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