10 Real Numbers, Equations, and Inequalities

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.

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

Identify Rational, Irrational, and Real Numbers

Identify Rational, Irrational, and Real Numbers (continued)

Identify Rational, Irrational, and Real Numbers

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.

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

Identify Rational, Irrational, and Real Numbers

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) 0.181818 … (b) 3.125 (c) 0.20220222022220… (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.

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

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

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.

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 < 2 15 Reverse symbol.

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 10 6 > Reverse symbol.

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.

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 6 + 5. 2 [8 + 3(11)] Multiply 3(11). 2 [8 + 33] Add 8 + 33. 2[41] Multiply 2 times 41. 82

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

Remove Parentheses Using the Distributive Property Example 6b

Remove Parentheses Using the Distributive Property

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