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Algebra 2
Chapter 1
Section 1.1
Expressions and Formulas
Review of Key Vocabulary
• Variables:
Symbols (letters) used to represent unknown quantities.
• Algebraic
Expressions: Expressions that contain at least one variable.
• Monomial:
An algebraic expression that is a number, variable, or product of a number and
one or more variables.
• Constants:
Monomials that contain no variables.
• Coefficient: The numerical factor of a monomial.
Review of Key Vocabulary
• Degree:
(of a monomial) is the sum of the exponents of its variables.
• Power:
An expression in the form of xn . The word is also used to refer to the exponent
itself.
• Polynomial: A monomial or a sum of monomials.
• Terms:
(of a polynomial) the monomials that make up a polynomial.
• Like
Monomials that can be combined. The have the same variables to the same powers.
Terms:
Review of Key Vocabulary
• Trinomial:
A polynomial that has three unlike terms.
• Binomal:
A polynomial that has two unlike terms.
• Formula:
A mathematical sentence that expresses the relationship between certain quantities.
Practice Problems – Evaluating Expressions
Evaluate each expression if 𝑥 = 4, 𝑦 = − 2, and 𝑧 = 3.5.
1. 𝑧 – 𝑥 + 𝑦
2. 𝑥 + (𝑦 – 1)3
3. 𝑥 + [3(𝑦 + 𝑧) – 𝑦]
4.
𝑥 2 −𝑦
𝑧+2.5
Practice Problems – Using Formulas
Simple interest is calculated using the formula 𝐼 = 𝑝𝑟𝑡, where p represents the principal in dollars, r
represents the annual interest rate, and t represents the time in years. Find the simple interest I given in
each set of values.
1. p = $1,800, r = 6%, t = 4 years
2. p = $31,000, r = 2 ½ %, t = 18 months
Section 1.2
Properties of Real Numbers
R = Reals
I = Irrationals
W = Wholes
Q = Rationals
Z = Integers
N = Naturals
Practice – Sets of Numbers
• Name the sets of numbers to which each number belongs:
5
6
2
−
3
– 43
– 23.3
Properties of Real Numbers
Property
Addition
Multiplication
𝑎+𝑏 =𝑏+𝑎
𝑎∗𝑏 =𝑏∗𝑎
𝑎 + 𝑏 + 𝑐 = 𝑎 + (𝑏 + 𝑐)
𝑎 ∗ 𝑏 ∗ 𝑐 = 𝑎 ∗ (𝑏 ∗ 𝑐)
Identity
𝑎+0=𝑎 =0+𝑎
𝑎∗1=𝑎 =1∗𝑎
Inverse
𝑎 + −𝑎 = 𝟎 = −𝑎 + 𝑎
Commutative
Associative
Distributive
1
1
=𝟏= ∗𝑎
𝑎
𝑎
𝑎 𝑏 + 𝑐 = 𝑎𝑏 + 𝑎𝑐 and 𝑏 + 𝑐 𝑎 = 𝑏𝑎 + 𝑐𝑎
𝐼𝑓 𝑎 ≠ 0, 𝑡ℎ𝑒𝑛 𝑎 ∗
Practice – Properties of Real Numbers
• Name the property illustrated by:
−8 + 8 + 15 = 0 + 15
• Identify the additive inverse and multiplicative inverse for – 7.
Practice – Simplifying Expressions
• Simplify:
3 4𝑥 − 2𝑦 − 2(3𝑥 + 𝑦)
• Simplify:
1
2
16 − 4𝑎
3
− (12 +
4
20𝑎)
Section 1.3
Solving Equations
Key Vocabulary
• Open Sentence: A mathematical sentence containing one or more variables.
• Equation:
A mathematical sentence stating two mathematical expressions are equal.
• Solution:
(of an open sentence) Each replacement of a number for a variable in an open
sentence that results in a true sentence.
Properties of Equality
Property
Symbols
Examples
Reflexive
For any real number a, a = a
−7 + 𝑛 = −7 + 𝑛
Symmetric
For all real numbers, a and b,
if a = b, then b = a
𝐼𝑓 3 = 5𝑥 + 6,
𝑡ℎ𝑒𝑛 5𝑥 + 6 = 3
Transitive
For all real numbers a, b, and c, if
a = b and b = c, then a = c.
𝐼𝑓 2𝑥 + 1 = 7 𝑎𝑛𝑑 7 = 5𝑥 − 8,
𝑡ℎ𝑒𝑛 2𝑥 + 1 = 5𝑥 − 8.
Substitution
If a = b, then a may be replaced
by b and b may be replaced by a.
𝐼𝑓 4 + 5 𝑚 = 18,
𝑡ℎ𝑒𝑛 9𝑚 = 18.
Practice – Algebraic to Verbal Sentence
• Write a verbal sentence to represent each equation:
𝑔 − 5 = −2
2𝑐 = 𝑐 2 − 4
Practice – Properties of Equality
• Name the property illustrated by each statement:
1. 𝐼𝑓 − 11𝑎 + 2 = −3𝑎, 𝑡ℎ𝑒𝑛 − 3𝑎 = −11𝑎 + 2.
2. 𝑎 − 2.03 = 𝑎 − 2.03
Tips to Remember When Solving Equations…
• Goal of solving an equation:
• Get the variable alone on one side of the equation and everything else on the other side.
• What you do to one side of the equation, you MUST do to the other side.
• Checking solutions to discover possible errors is a vital procedure when you use math on the
job.
• Use reverse-PEMDAS when solving multi-step equations.
Practice – Solving Equations
• Solve:
𝑥 − 14.29 = 25
• Solve :
2
𝑦
3
= −18
Practice – Solving Equations
• Solve:
−10𝑥 + 3 4𝑥 − 2 = 6
• Solve :
2 2x − 1 − 4 3x + 1 = 2
Practice – Solving Equations
• If 5y + 2 =
8
,
3
what is the value of 5𝑦 − 6?
Practice – Solving Equations
The formula for the surface area S of a cylinder is S = 2π𝑟 2 + 2𝜋𝑟ℎ, where
𝑟 is the radius of the base, and ℎ is the height of the cylinder.
• Solve the formula for ℎ
Section 1.5
Solving Inequalities
Trichotomy Property
• For any two real numbers, 𝑎 and 𝑏, exactly one of the following statements
are true:
𝑎<𝑏
𝑎=𝑏
𝑎>𝑏
Adding the same number to, or subtracting the same number from, each side of an inequality does
NOT change the truth of the inequality.
Properties of Inequality
Addition Property of Inequality
Words
Example
For any real numbers, 𝑎, 𝑏, and 𝑐:
3<5
3 + −4 < 5 + −4
If 𝑎 > 𝑏, then 𝑎 + 𝑐 > 𝑏 + 𝑐.
If 𝑎 < 𝑏, then 𝑎 + 𝑐 < 𝑏 + 𝑐.
−1 < 1
Subtraction Property of Inequality
Words
For any real numbers, 𝑎, 𝑏, and 𝑐:
If 𝑎 > 𝑏, then 𝑎 − 𝑐 > 𝑏 − 𝑐.
If 𝑎 < 𝑏, then 𝑎 − 𝑐 < 𝑏 − 𝑐.
Example
2>7
2−8>7−8
−6 > −15
Practice – Solve an Inequality Using Addition
• Solve 4𝑥 + 7 ≤ 3𝑥 + 9. Graph the solution set on a number line.
Multiplication Property of Inequality
Words
Examples
For any real numbers, a, b, and c, where:
If 𝑎 > 𝑏, the 𝑎𝑐 > 𝑏𝑐
c is positive:
If 𝑎 < 𝑏, then 𝑎𝑐 < 𝑏𝑐
−2 < 3
4 −2 < 4 3
−8 < 12
If 𝑎 > 𝑏, then 𝑎𝑐 < 𝑏𝑐
c is negative:
If 𝑎 < 𝑏, then 𝑎𝑐 > 𝑏𝑐
5 > −1
−3 5 < −3 −1
−15 < 3
Division Property of Inequality
Words
Examples
For any real numbers, a, b, and c, where:
𝑎
If 𝑎 > 𝑏, then 𝑐 >
c is positive:
𝑎
If 𝑎 < 𝑏, then 𝑐 <
𝑏
𝑐
𝑏
𝑐
−18 < −9
−18 −9
<
3
3
−6 < −3
𝑎
If 𝑎 > 𝑏, then 𝑐 <
c is negative:
𝑎
If 𝑎 < 𝑏, then 𝑐 >
𝑏
𝑐
𝑏
𝑐
12 > 8
12
8
<
−2 −2
−6 < −4
Set-builder Notation
• 𝑥 𝑥 > 9 is read
“The set of all numbers x such that x is greater than 9.”
{ } (called braces) denotes “the set of ”
| denotes “such that”
Practice – Solve an Inequality Using
Multiplication
• Solve
1
− 𝑥
3
< 1. Graph the solution set on a number line.
Solve a Multi-Step Inequality
• Solve 3 2𝑞 − 4 > 6. Graph the solution set on a number line.
Solve a Multi-Step Inequality
• Solve −𝑥 >
𝑥−7
.
2
Graph the solution set on a number line.