Download 1.3 Operations with Real Numbers (Cont.)

Intermediate Algebra
By Alan S. Tussy
5th Edition
Instructor: Valerie Merrill
Objectives:
At the end of this lesson, the student
will be able to evaluate, solve and
create equations and expressions
using real numbers.
Student Learning Outcomes:
To be prepared to continue in the
study of Algebra at the Intermediate
level.
An algebraic expression is a combination of variables (a letter used to represent
a number) and/or numbers with the operations (addition, subtraction,
multiplication, division, powers, roots,…).
An equation is a mathematical sentence that contains an = symbol.
This indicates the expressions on either side of have the same value.
Some common math vocabulary:
Addition - add, sum, plus, increased by, greater than, more than, total…
Subtraction – subtract, difference, less than, decreased by, minus, reduced
by, subtracted from,…
*watch the order for “less than” and “subtracted from”*
Multiplication – multiplied by, product, percent or fraction of, times, twice,
three times,…
Division - divided by, quotient of, ratio, half, into, per,…
Equal Sign - equals, the same as, is, gives, yields,…
Note: Tables, bar graphs, line graphs and mathematical models are all
ways to model information in order to determine relationships.
Natural numbers
ℕ - {1, 2, 3, 4, 5, …}
Whole numbers
𝕎 – {0, 1, 2, 3, 4, …}
Integers
ℤ - {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}
𝒂
Rational Numbers
ℚ - {all numbers that can be written as fractions 𝒃
where a and b are integers and 𝑏 ≠ 0}
Note: Rational numbers include all terminating and repeating
decimals.
Irrational Numbers
ℍ - {all numbers that can not be expressed as a
fraction with integer numerator and nonzero
integer denominator}
Note: Irrational numbers consist of nonrepeating,
nonterminating decimals. ( 2, 𝜋, 0.121221222 … )
Real numbers
ℝ - {all rational and irrational numbers}
Some symbols used to compare values are:
≠ means not equal to
< means is less than
≤ means is less than or equal to
> means is greater than
≥ means is greater than or equal to
All real numbers correspond to a point on the number line.
Two numbers that are the same distance from zero but are on opposite sides
of zero on the number line are called opposites.
a and –a are opposites
-a and –(-a) are opposites
The absolute value of a number is the distance the number is from zero on
the number line.
For any real number x
If 𝒙 ≥ 𝟎, then 𝒙 = 𝒙
If 𝒙 < 𝟎, then 𝒙 = −𝒙
Some symbols used in set notation are:
∈
which indicates that an element belongs to a set
𝟏
−
𝟐
⊆
∈ℚ
𝟑∈ℍ
which indicates that one set is a subset of another set,
meaning that all the members of one set are members of
the other set as well
ℕ⊆𝕎
ℚ⊆ℝ
Note: The symbols ⊈ and ⊉ can be used to state that a set is not a
subset of another as well.
To add numbers with the same sign, add their absolute values and the sum
has the same sign as the original numbers.
𝟏
𝟐
𝟓. 𝟑 + 𝟐. 𝟗 = 𝟖. 𝟐
− + −
𝟏
𝟒
=−
𝟑
𝟒
To add two numbers with difference signs, subtract the absolute value of
the smaller number from the larger and the sum has the sign of
the number with the larger absolute value.
𝟑
− 𝟖 + 𝟕 = 𝟔. 𝟔𝟐𝟓
𝟏
𝟑
𝟓
To subtract two numbers, add the opposite.
𝒂 − 𝒃 = 𝒂 + (−𝒃)
−𝟗 − 𝟑 = −𝟏𝟐
𝟐
+ −𝟗 = −𝟗
𝟒 − 𝟔 = −𝟐
The product or quotient of two numbers with the same sign is positive.
+ ∙ += +
− ∙ −= +
+ ÷ += +
− ÷ −= +
The product or quotient of two numbers with different signs is negative.
+ ∙ −= −
− ∙ += −
𝒙𝒏 means 𝒙 ∙ 𝒙 ∙ 𝒙 ∙ 𝒙 ∙ 𝒙 ∙ ⋯ ∙ 𝒙
+ ÷ −=−
− ÷ += −
n factors of x
b is a square root of a iff 𝒃𝟐 = 𝒂.
𝒂 asks for the principal square root of a.
𝑥
𝑛
base
exponent
Order of Operations:
1. Parentheses (from innermost to outermost grouping symbols)
2. Exponents
3. Multiplication/Division (from left to right)
4. Addition/Subtraction (from left to right)
Example.
Evaluate.
7−6
36 −
4
+ 32
16 + 1
2
To evaluate and algebraic expression means to substitute the values for the
variables and simplify using the order of operations.
Example.
Evaluate.
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
for
𝑎 = 2, 𝑏 = −3, and 𝑐 = −2
A term is a product or quotient of numbers and/or variables. A single
number term is called a constant.
𝒙
− 𝟑𝒙𝟐 𝒚
𝟏
𝒂𝒃
𝟐
− 𝟏. 𝟔𝒂𝟑 𝒃𝒄𝟒
−𝟐
The numerical factor of a term is called the coefficient.
Terms with the exact same variables raised to the exact same
exponents are called like terms. To combine like terms add or
subtract the coefficients and keep the variables the same with
the same exponents as well.
Example.
Simplify.
3
5
2ℎ2 + 9ℎ − ℎ2 − ℎ
4
4
Properties of Real Numbers:
Commutative Properties
of addition
of multiplication
Associative Properties
of addition
of multiplication
Additive Identity (Zero)
Multiplicative Identity (One)
Multiplicative Property of Zero
Additive Inverse Property
Multiplicative Inverse Property
Division Properties
𝒂
𝟎
𝒂+𝒃=𝒃+𝒂
𝒂∙𝒃= 𝒃∙𝒂
𝒂+𝒃 +𝒄=𝒂+ 𝒃+𝒄
𝒂∙𝒃 ∙𝒄=𝒂∙ 𝒃∙𝒄
𝒂+𝟎=𝟎+𝒂= 𝒂
𝒂∙𝟏= 𝟏∙𝒂=𝒂
𝒂∙𝟎= 𝟎∙𝒂=𝟎
𝒂 + −𝒂 = −𝒂 + 𝒂 = 𝟎
𝟏
𝟏
𝒂∙𝒂= 𝒂∙𝒂=𝟏
𝒂
𝟏
=𝒂
𝐢𝐬 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
𝒂
𝒂
=𝟏
𝟎
𝟎
𝟎
𝒂
=𝟎
𝐢𝐬 𝒊𝒏𝒅𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒕𝒆
Distributive Property
over addition
𝒂 𝒃 + 𝒄 = 𝒂𝒃 + 𝒂𝒄
𝒃 + 𝒄 𝒂 = 𝒃𝒂 + 𝒄𝒂
over subtraction𝒂 𝒃 − 𝒄 = 𝒂𝒃 − 𝒂𝒄
𝒃 − 𝒄 𝒂 = 𝒃𝒂 − 𝒄𝒂
extended 𝒂 𝒃 + 𝒄 + 𝒅 + 𝒆 + ⋯ = 𝒂𝒃 + 𝒂𝒄 + 𝒂𝒅 + 𝒂𝒆 + ⋯
A linear equation in one variable is an equation that can be written in the
form 𝒂𝒙 + 𝒃 = 𝒄 where 𝒂 ≠ 𝟎.
A solution to an equation is a number that when substituted in for the
variable makes a true statement.
Using the following properties you can solve linear equations.
Addition/Subtraction Property of Equality
If 𝒂 = 𝒃, then
𝒂+𝒄=𝒃+𝒄
𝒂 − 𝒄 = 𝒃 − 𝒄.
Multiplication/Division Property of Equality
If 𝒂 = 𝒃, then
𝒂 ∙ 𝒄 = 𝒃 ∙ 𝒄 for 𝒄 ≠ 𝟎
𝒂
𝒄
=
𝒃
𝒄
for 𝒄 ≠ 𝟎.
To solve linear equations:
1. Clear any fractions (multiply both sides by the LCD)
or decimals (multiply both sides by a power of 10).
2. Distribute to clear all parentheses and combine any like terms.
3. Move all variables to one side of the equation.
Use the addition/subtraction property of equality.
4. Undo any addition/subtraction on the variable.
Use the addition/subtraction property of equality to perform
inverse operations.
5. Undo any multiplication/division on the variable.
Use the multiplication/division property of equality to
perform inverse operations.
6. Check all solutions in the original equation.
If the variables cancel out of the equation leaving:
𝟏
𝟏
- A true statement (𝟎 = 𝟎, 𝟓 = 𝟓, − = − , … ) then the equation is
𝟐
𝟐
true for all values of the variable and it is called an identity.
𝟑
- A false statement (𝟎 = 𝟏, 𝟓 = −𝟓, 𝟒 = 𝟎. 𝟐𝟓, … ) then the equation is
true for no values of the variable and it is called a
contradiction.
Examples.
Solve.
𝑎+3
+
2
3
2𝑎 = 2 −
𝑎+27
5
𝑦 + 0.5 = 2.5 0.2𝑦 + 1 − 0.5 4 − 𝑦
−3𝑥 = −2𝑥 + 1 − (5 + 𝑥)
Formulas state relationships between two or more variables.
Note: Some common geometric formulas are on the back inside
cover of your textbook.
To solve a formula for a specific variable, isolate that variable using the steps
in solving linear equations.
Example.
Solve for M.
𝑀𝑣0 2 + 𝑙𝑤 2
𝑘=
2
To solve real-life word problems:
1. Read the problem carefully.
Identify the knowns and unknowns, choose a variable,
locate key terms, draw a picture/chart/table/diagram.
2. Form an equation.
Use formulas or key terms to write and equation.
3. Solve the equation.
Use the steps from to solve linear equations.
4. Answer the problem.
Label all solutions.
5. Check the result.
Make sure that the answer makes sense and solves the
equation formed.
Example.
Solve.
A 186 foot board is cut into four pieces where each piece is 3 feet longer
than the last. Find all the lengths of the boards.
Some common formulas used in problem solving are in the form of:
Percent –
∎ is ∎% of ∎?
translates to
∎=∎∙∎
or
as a decimal
𝒊𝒔
𝒐𝒇
%
= 𝟏𝟎𝟎
Note: For percent increase or decrease, the of is always the
original amount and the is always represents the
change in value.
Statistics –
Mean =
𝐬𝐮𝐦 𝐨𝐟 𝐭𝐡𝐞 𝐯𝐚𝐥𝐮𝐞𝐬
𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐭𝐡𝐞 𝐯𝐚𝐥𝐮𝐞𝐬
Median = the middle number when values are places in
ascending order
Mode = the value that occurs most often
Investment –
𝑰 = 𝑷𝒓𝒕 represents I – Interest, P-principal, r-rate (as a
decimal), t-time (in years)
Uniform Motion –
𝒅 = 𝒓𝒕
represents
d-distance, r-rate, t-time
Mixture –
Amount ∙ Price = Value
Amount ∙ Strength = Pure Ingredient
(For dry mixtures)
(For liquid mixtures)
The following chart can be useful with problem solving:
Type 1
Type 2
Total
Case 1
∙
=
Case 2
∙
=
Form equations from here
Examples.
Solve.
Lindsey Lohan leaves a club traveling 60 mph. One minute later a
motorcycle cop leaves the club on his motorcycle traveling 90 mph.
How long before the cop catches Lindsey?
How much of a 4% pesticide solution must be added to 20 gallons of a 12%
pesticide solution to dilute it to a 10% solution?