Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
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California StandardsCalifornia Standards
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Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
Warm UpWrite each fraction as a decimal.
1. 1 3
0.3 2. 45
3. 4.
0.8
0.753 4
23
0.6
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
NS1.1 Read, write, and compare rational numbers in scientific notation (positive and negative powers of 10), compare rational numbers in general.Also covered: NS1.3
California Standards
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
least common denominator (LCD)
Vocabulary
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
To compare fractions with unlike denominators, you can find a common denominator. This could be the least common denominator (LCD), which is the least common multiple of the denominators.
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational NumbersAdditional Example 1A: Comparing Fractions by Finding a
Common DenominatorCompare. Write <, >, or =.
Multiply 6 and 10 to find a common denominator.
5 6
7 10
Write the fractions with a common denominator.
Compare the fractions.
Method 1: Multiply to find a common denominator.
6 10 = 60
5 10 6 10
=5 6
10 10
= 50 60
7 6 10 6
=7 10
6 6
= 42 60
50 60
5 6
> 42 60
,so 7 10
>
>
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
Remember!
The least common multiple (LCM) of two numbers is the smallest number, other than 0, that is a multiple of both numbers.
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational NumbersAdditional Example 1B: Comparing Fractions by Finding a
Common DenominatorCompare. Write <, >, or =.
List multiples of 3 and 5. The LCM is 15.
2 3
4 5
Write the fractions with a common denominator.
Compare the fractions.
Method 2: Find the least common denominator.
3; 3, 6, 9, 12, 15… 5; 5, 10, 15
2 5 3 5
=2 3
5 5
= 10 15
4 3 5
3
=4 5
3 3
= 12 15
10 15
2 3
< 12 15
,so 4 5
<
>
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
Check It Out! Example 1A
Compare. Write <, >, or =.
Multiply 2 and 5 to find a common denominator.
1 2
2 5
Write the fractions with a common denominator.
Compare the fractions.
Method 1: Multiply to find a common denominator.
2 5 = 10
1 5 2 5
=1 2
5 5
= 5 10
2 2 5 2
=2 5
2 2
= 4 10
5 10
1 2
> 4 10
,so 2 5
>
>
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
Check It Out! Example 1BCompare. Write <, >, or =.
List multiples of 3 and 4. The LCM is 12.
2 3
3 4
Write the fractions with a common denominator.
Compare the fractions.
Method 2: Find the least common denominator.
3; 3, 6, 9, 12, … 4; 4, 8, 12…
2 4 3 4
=2 3
4 4
= 8 12
3 3 4
3
=3 4
3 3
= 9 12
8 12
2 3
< 9 12
,so 3 4
<
>
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
A. 5 5
Additional Example 2: Comparing by Using Decimals
Compare. Write <, >, or =.
2 9 _ 2
7 _
Write the fractions as decimals.
5 = 5.2 and 5 = 5.285714… 2 9 _ 2
7 __
5.2 < 5.285714…, so 5 < 5 _ 2
9 _ 2
7 _ Compare the
decimals.
<
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
B. –0.44 –
Additional Example 2: Comparing by Using Decimals
Compare. Write <, >, or =.2 5 _
– = –0.4 2 5 _
–0.44 < –0.4, so –0.44 < –2 5 _ Compare the
decimals.
<
Write - as decimal.2 5 _
C. 0.1 1 9 _
Compare the decimals.
Write as decimal.1 9 _
>
1 9 _ = 0.1
0.1 > 0.1, so > 0.11 9
_
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
A. 4 4
Check It Out! Example 2
Compare. Write <, >, or =.
2 9 _ 3
5 _
Write the fractions as decimals.
4 = 4.2 and 4 = 4.6 2 9 _ 3
5 __
Compare the decimals.
<
4.2 < 4.6, so 4 < 4 2 9 _ 3
5 _
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
B. –0.80 –
Check It Out! Example 2
Compare. Write <, >, or =.
4 5 _
– = –0.8 4 5 _
–0.80 = –0.8, so –0.80 = –4 5 _ Compare the
decimals.
=
Write – as decimal.4 5 _
C. 0.8 5 6 _
Compare the decimals.
Write as decimal.5 6 _
>
0.83 > 0.8, so > 0.85 6
_ _
5 6 _ = 0.83
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
To order fractions and decimals, you can either write them all in the same form and then compare them, or place them on a number line. Recall that numbers increase in value as you move from left to right along a number line.
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational Numbers
Additional Example 3: Social Studies Application
The numbers , –3.4, 6.0, and –2.5
represent the percent changes in populations for four states. List these numbers in order from least to greatest.
14 4__
–4 –3 –2 –1 0 1 2 3 4 5 6
–2.5
6.0
Place the numbers on a number line and read them from left to right.
–3.4
14 4__
14 4
The percent changes in population from least to greatest are –3.4, –2.5, , and 6.0.__
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational NumbersCheck It Out! Example 3
The numbers , 3.0, –2.2, and –3.9
represent the percent changes in populations for four states. List these numbers in order from least to greatest.
7 2__
–5 –4 –3 –2 –1 0 1 2 3 4 5
–2.2
3.0
Place the numbers on a number line and read them from left to right.
–3.9
7 2__
The percent changes in population from least to greatest are –3.9, –2.2, 3.0, and .7
2
Evaluating Algebraic Expressions
2-2 Comparing and Ordering Rational NumbersLesson Quiz
Compare. Write <, >, or =.
1.
2. – –0.29
4. Sarah competed in a long-jump contest. Her
first jump was 3.75 m, her second jump was 3 m,
and her third jump was 3 m. Which jump was the longest?
3. –2 –2 >
>
<1 3
second jump
1 4
2 9
6 7
7 8
8 99
11