1. 10
Real Numbers, Equations, and Inequalities

2. 10.1 Real Numbers and Expressions
Objectives
Identify rational numbers, irrational numbers, and real numbers.
Use the symbols ≠, <, ≤, >, and ≥ to compare real numbers.
Reverse the direction of inequality statements.
Use the order of operations to simplify expressions with brackets.
Remove parentheses and simplify expressions using the distributive property.

3. Identify Rational, Irrational, and Real Numbers
Familiar types of numbers:
natural numbers
whole numbers
integers

4. Identify Rational, Irrational, and Real Numbers

5. Identify Rational, Irrational, and Real Numbers
(continued)

6. Identify Rational, Irrational, and Real Numbers

7. Graphing Rational Numbers
Example 1 Graph each number on the number line.
To locate the improper fractions on the number line,
write them as mixed numbers or decimals.

8. Identify Rational, Irrational, and Real Numbers
There are numbers that are not rational.
The pattern never repeats and never ends. π is irrational.

9. Identify Rational, Irrational, and Real Numbers

10. Identify Rational, Irrational, and Real Numbers
Example 2
Identify each number as rational or irrational, and explain why. Use your calculator to find square roots.
(a) … (b) (c) …
(d) (e) (f )
(a) Rational, because the digits repeat in a fixed block.
(b) Rational, because the decimal terminates (comes to an end).
(c) Irrational, because the digits do not repeat in a fixed block.
(d) Irrational, because the decimal value never terminates
or repeats.
(e) Rational, because simplified it equals 4.
(f ) Rational, because the digits repeat in a fixed block.

11. Identify Rational, Irrational, and Real Numbers
All numbers that can be represented by points on the number line are called real numbers.

12. Use ≠, <, ≤, >, and ≥ to Compare Real Numbers

13. Inequalities
Example 3 #
Example
True or False?
(a)
6 ≠ 1
True
(b)
9 ≥ 5
True
(c)
8 < 4
False
(d)
1 > 2
False
(e)
6 ≤ 6
True
If either the < part or the = part is true, then the inequality ≤ is true.

14. Converting Inequalities
To convert between < and >, reverse both the order of the numbers and the direction of the symbol.
Example 4
Interchange numbers.
15 > 2
becomes
<
Reverse symbol.

15. Converting Inequalities
To convert between < and >, reverse both the order of the numbers and the direction of the symbol.
Example 4
Interchange numbers.
6 < 10
becomes
>
Reverse symbol.

16. Using the Order of Operations to Simplify Expressions
We have been using parentheses to show several different things.
An expression with double parentheses, such as 2(8 + 3(6 + 5)), can be confusing. Use square brackets [ ] in place of one set of parentheses.

17. Using the Order of Operations to Simplify Expressions
Example 5
Simplify. 2[8 + 3(6 + 5)]
Begin inside the parentheses. Then follow the order of operations as you complete the work inside the brackets.
2 [8 + 3(6 + 5)] Work inside parentheses: add
2 [8 + 3(11)] Multiply 3(11).
2 [8 + 33] Add
2[41] Multiply 2 times 41. 82

18. Remove Parentheses Using the Distributive Property
Example 6a
Write without parentheses.
(a)

19. Remove Parentheses Using the Distributive Property
Example 6b

20. Remove Parentheses Using the Distributive Property

21. Remove Parentheses Using the Distributive Property
Simplify: 5(2a2 – 6a) – 3(4a2 – 9)
Example 8