Section 1.1 Real Numbers and Number Operations - Edl · Section 1.1 Real Numbers and Number Operations ... Properties of Addition and Multiplication ... 2. Exponents (including square

Reflection: 1

Section 1.1 Real Numbers and Number Operations Objective(s): Differentiate among subsets of the real number system.

Essential Question: What is the difference between a rational and irrational number?

Homework: Assignment 1.1. #1 – 14 in the homework packet.

You MUST show all work on a SEPARATE sheet of paper.

As a heading, include name, date, period, teacher’s name, and title (Homework Section 1.1).

Notes:

Vocabulary

The opposite of a number x is –x.

If a is not 0, then a and 1

a are called reciprocals.

Properties of Addition and Multiplication

Property Addition Multiplication

Commutative a + b = b + a ab = ba

Associative a + (b + c) = (a + b) + c a(bc) = (ab)c

Distributive a(b + c) = ab + bc

Sets of Numbers

Set of Numbers Examples Notation

Natural Numbers 1, 2, 3, 4, … Whole Numbers 0, 1, 2, 3, …

Integers …, -3, -2, -1, 0, 1, 2, 3, … Rational Numbers Numbers that can be written as

fractions (also includes decimals and repeating decimals)

Irrational Numbers Numbers that cannot be written as

fractions (square roots, , nonrepeating decimals, etc.)

Real Numbers Includes all of the above

Reflection: 2

Tell which set or sets the number belongs to: natural numbers, whole numbers, integers, rational

numbers, irrational numbers, and real numbers.

Circle ALL correct choices.

Example 1: 3 natural, whole, integer, rational, irrational, real


Example 3: -7 natural, whole, integer, rational, irrational, real

Example 4: 0.123123123… natural, whole, integer, rational, irrational, real



Write a sentence using mathematical symbols.

Example 7: The difference of 3 times x and 4 is greater than 12. _________________________

Example 8: Five times x is 32 more than twice x. _________________________

Example 9: The quotient of 18 and x equals 6. _________________________

Example 10: The sum of twice x and 7 is 19. _________________________

Example 11: Twice the sum of x and 7 is 19. _________________________

Example 12: Nine subtracted from 4 times x is equal to 100. _________________________

Real Numbers

Irrational Numbers Rational Numbers

Integers

Whole

Numbers Natural

Reflection: 3

Section 1.2 Algebraic Expressions and Models Objective(s): Simplify numerical and algebraic expressions applying the appropriate field properties.

Essential Question: Describe what it means for terms to be like terms and give an example.




Notes:

Vocabulary

A letter that represents a number is called a variable.

An exponent is a shorthand notation for repeated multiplication of the same factor.

An algebraic expression is formed by numbers and variables connected by the operations of addition,

subtraction, multiplication, division, raising to powers, and/or taking roots.

Order of Operations

1. Grouping symbols (including parentheses, brackets [ ], braces { }, absolute value | |, and the

fraction bar)

2. Exponents (including square roots, cube roots, etc.)

3. Multiplication and Division (left to right)

4. Addition and Subtraction (left to right)

Note: When simplifying a numeric expression, parentheses, brackets, and braces all mean the same

thing. Work from the inside out. This usually means do the parentheses, and then the brackets, and then

the braces.

Mnemonic device for the Order of Operations: PEMDAS or “Please Excuse My Dear Aunt Sally”

Note: Be careful when using this mnemonic device. Remember that multiplication/division and

addition/subtraction are performed left to right!

Evaluate the expression 22 6 3 2 5x x y when 3x and 5y .

Substitute values for x and y: 2

2 3 6 3 3 5 2 5

Reflection: 4

Grouping symbols:

2

2

2 3 6 3 15 2 5

2 3 6 12 2 5

Exponents: 2 9 6 12 2 5

Mult/Div Left to Right: 18 72 2 5

18 36 5

Add/Sub Left to Right: 18 5

13

Simplify the expression.

Example 1: 18 2 6 What do you do first? _____________________

What do you do second? __________________

Answer: ________________________________

Example 2: 24 2 5 What do you do first? _____________________


What do you do third? ____________________

Answer: ________________________________

Example 3: 1 1 1

6 3 4

Answer: ________________________________

Example 4: 3 7 2 8 3 What do you do first? _____________________


What do you do third? ____________________

Answer: ________________________________

Reflection: 5

Evaluate the expression for the given replacement values.

Example 5: 14 9

5

x y

x for x = 7 and y = 5

Answer: ________________________________

Example 6: (x + 3y)2 for x = 2 and y = 4

Answer: ________________________________

Example 7: -4(3x + 8y) for x = 2 and y = 1/8

Answer: ________________________________

Simplifying Algebraic Expressions

To simplify an algebraic expression, eliminate parentheses using the distributive property and combine

like terms. Terms are separated by (+) or (–) signs.

What is the simplified form of the expression?

Example 8: 5 + (2k + 6)

Example 9: -5m + 4 – 6 + 4 + m – 2

Example 10: 6(5m – 3) – 2(3m + 6)

Reflection: 6

Sample CCSD Common Exam Practice Question(s):

1. To which sets of numbers does 3

5 belong?

I. integers

II. natural numbers

III. rational numbers

IV. real numbers

V. whole numbers

A. I, III, and IV only

B. II and IV only

C. III and IV only

D. III, IV, and V only

2. Evaluate (3 )c d when 1c and 5d .

A. 8

B. 2

C. 3

D. 8

3. Which is a simplified form of the expression 3( 2) 5(2 4)x x ?

A. 16x

B. 19x

C. 7 2x

D. 7 26x

Reflection: 7

Section 1.3 Solving Linear Equations Objective(s): Solve linear equations in one variable.

Essential Question: What role does the order of operations play when solving an equation?




Notes:

To solve an equation, we use the properties of equality to isolate the variable on one side of the

equation.

When isolating the variable, we will “undo” addition/subtraction first, then multiplication/division. This

is the order of operations in reverse!

Ex: Solve the equation 8 6 5x .

Step One: Subtract 6 from both sides. 8 1x

Step Two: Divide both sides by 8 . 1

8x

For more complicated equations, you may have to simplify both sides of the equation first using the

distributive property and combining like terms.

Ex: Solve the equation 6 4 2 24x x

Step One: Use the distributive property. 6 24 2 24x x

Step Two: Combine like terms. 4 24 24x

Step Three: Add 24 to both sides. 4 48x

Step Four: Divide both sides by 4. 12x

Reflection: 8

If there are variables on both sides, you will need to simplify both sides of the equation and then bring

all variables to the same side using inverse operations.

Ex: Solve the equation 3 2 4 6x x

Step One: Use the distributive property. 3 2 4 24x x

Step Two: Subtract 3x from both sides. 2 24x

Step Three: Subtract 24 from both sides. 26 x

Note: We can also write the answer in set notation. The solution set to the equation above is 26 .

When solving an equation with fractions, it saves time to “wipe-out” the fractions by multiplying both

sides by the LCM (least common multiple).

Ex: Solve the equation: 3 9 1

25 10 3

x x

The LCM of the fractions is 30.

“Multiply” every term (NUMERATOR ONLY) in the equation by 30 – but don’t actually multiply

because you cancel first.

3 9 1

30 30 30 30 25 10 3

x x

3 9 130 30 30 30 2

5 10 3x x

6 3 3 9 10 1 30 2x x

18 27 10 60

27 10 42

37 42

37

42

x x

x

x

x

Solve the equation.

Example 1: -5n = -15

Example 2: 10r + 12 = 62

Reflection: 9

Example 3: 6(3x – 1) = 24 What do you have to do first? ______________________

Example 4: -2x – 7x + 6 = 2x

Example 5: 1

3 17

x What do you have to multiply the numerators by? ______

Example 6: 5 1 7

3 2 6x What do you have to multiply the numerators by? ______

Example 7: 1 1 4

23 6 3

x x


1. What is the value of x when 2 1 5

3 2 6x ?

A. 2

9x C.

5

7x

B. 1

2x D.

3

2x

Reflection: 10

Section 1.4 Rewriting Equations and Formulas Objective(s): Solve for a given variable in an equation with more than one variable.

Essential Question: Describe a real-life situation where you would need to be able to solve the area

formula of a circle for r.




Notes:

The equation 3 2 5x y is an example of an equation that is implicitly defined. Sometimes it is useful

to rewrite the equation so that it is explicitly defined, which means it is solved for one of the variables

(almost always solving for y).

Ex: Solve the equation 3 2 5x y for y .

Step One: Subtract 3x from both sides. 2 5 3y x

Step Two: Divide both sides by 2. 5 3

2 2y x

Note: There are many ways to write the correct answer. This equation is equivalent to the

following:

3 5

2 2y x (most typical) and

5 3

2

xy (hardly ever)

Sometimes it is useful to solve formulas for one of the variables.

Ex: Solve the formula for the area of a trapezoid for 1b .

1 2

1

2A h b b

Step One: Multiply both sides by 2 (LCD). 1 22A h b b

Step Two: Use the distributive property. 1 22A hb hb

Reflection: 11

Step Three: Subtract 2hb from both sides. 2 12A hb hb

Step Four: Divide both sides by h . 21

2A hbb

h or 2 1

2Ab b

h

Note: LOOK for opposites. These questions will always be multiple choice. If the original problem has

addition, then the answer MUST have subtraction. If the original problem has division, then the

answer MUST have multiplication, etc.

Solve the formula for the specified variable.

Example 1: E = mc2 for c

A. 2

Emc

B. 2

Ec

m

C. E

cm

D. c Em

Example 2: 21

3V r h for r

A. 3V

rh

B. 3V

rh

C. 3

Vr

h

D. 3

Vr

h


1. Which represents y in terms of x for the equation 2 5 6 10 8x y x x y ?

A. 1

22

y x B. 3

22

y x C. 7

22

y x D. 9

22

y x

Reflection: 12

Section 1.5 Problem Solving Using Algebraic Models Objective(s): Develop a mathematical model to solve real-world equations.

Essential Question: List and describe the steps of a problem solving plan.




Notes:

When you have a real world (word problem) that requires you to write an equation in slope-intercept form, there are two things that you want to look for:

Example 1: You are visiting a Baltimore MD, and a taxi company charges a flat fee of $3.00 for using

the taxi and an additional $0.75 per mile. Let C represent the total cost and x represent the number of

miles. Write a model for the total cost of the taxi ride.

What is the flat fee? __________________________

What is the rate? _________

Model ____________________________________

Reflection: 13

Example 2: In 1960, the average life-span of an American woman was 73 years. Every year (since

1960), the life-span for American women has increased by 0.2 years. Let A represent the average life-

span and t represent the number of years since 1960. Write a model for the average life-span of an

American woman.

Example 3: A plumber charges a fee of $50 to make a house call. He also charges $25 per hour for

labor. Let C represent the total cost and h represent the number of hours of the house call. Write a

model for the total cost of the plumbers house call.


1. The cost for parts to repair a car was $1,015. The cost for labor was $70 per hour. Which of

the following is a linear model for the total cost of repairing the car where n represents the

number hours of labor?

A. 70 1015C n

B. 1015 70C n

C. 70 1015C n

D. 1015 70C n

Reflection: 14

Section 1.6 Solving Linear Inequalities Objective(s): Solve linear inequalities in one variable.

Essential Question: When solving an inequality, when must you flip the sign, and why?




Notes:

Inequality Symbols:

Greater Than Greater Than or Equal To

Less Than Less Than or Equal To

Graphing an Inequality: Use parentheses for greater than (>) or less than (<). Use a bracket for

greater than or equal to ( ) or less than or equal to ( ).

Ex: Graph the inequality 8x

Graph the inequality on a number line.

Example 1: x 3

Example 2: x > 1

Example 3: -8 < x -4

Example 4: 0 < x < 7

Interval notation is a notation for representing an interval as a pair of numbers. The numbers are the endpoints of the interval. Parentheses and/or brackets are used to show whether the endpoints are excluded or included. For example, [3, 8) is the interval of real numbers between 3 and 8, including 3 and excluding 8.

Write examples #1 – 4 in interval notation.

Example 5: x 3

Example 6: x > 1

Example 7: -8 < x -4

Example 8: 0 < x < 7

) 0 1 2 3 4 5 6 7 8 9

Reflection: 15

Graph the set of numbers given in interval notation.

Example 9: ( -3, ∞)

Example 10: ( 9, 12]

Solving a Linear Inequality: solving an inequality is similar to solving an equation, with one exception.

*When multiplying or dividing by a negative number in an inequality, you must FLIP the inequality sign*

Ex: Solve the inequality 4x .

To isolate the variable, we subtract 4 from both sides and add x to both sides to obtain 4 x .

Which, written with the variable on the left would read 4x .

(ALWAYS write your answer with the variable on LEFT.)

An alternative way to solve this inequality would be to divide both sides by 1 .

4

1 1

x Note: We needed to FLIP (reverse) the inequality symbol!

Solve the inequality. Graph the solution set. Then write the solution in interval notation.

Example 11: 9x – 11 > 8x – 14

Graph:

Interval Notation: _____________________

Example 12: 1

36

x

Graph:


Example 13: -5(4x + 13) < -25x – 35

Graph:


Reflection: 16

Compound Inequalities are two inequalities joined by “and” or “or”

Ex: 5 or 2x x

Ex: 3 and 4x x Note: This can also be written as 3 4x

Solving a compound inequality

Isolate the variable. In an “and” statement, isolate the variable between the two inequality signs. What

you do to the middle, you must do to all sides!

Ex: Solve and graph: 3 1 2 5x

Step One: Add 1 to the middle and both sides 2 2 6x

Step Two: Divide the middle and both sides by 2 1 6x Don’t forget to FLIP!

Step Three: Rewrite the inequality (smallest number on the left) 6 1x

Step Four: Graph on a number line or write in interval notation

Ex: Solve and graph: 6 5 7 or 8 1 25x x

Step One: Solve both inequalities separately 6 12x 8 24x

2x 3x

Step Two: Write the solutions as a compound inequality 2 or 3x x

Step Three: Graph on a number line or write in interval notation.

Solve the inequality. Graph the solution set and write it in interval notation.

Example 14: 5 3 4 14x

Graph:


Example 15: 9x – 6 < 3x or -2x -6

Graph:

Reflection: 17


Example 16: -3x + 1 7 or -5x + 3 -17

Graph:



1. Which expresses all of the solutions for the compound inequality below?

2 3 5 or 3( 8) 3x x

A. 1 9x

B. x = 1 or x = 9

C. 1x or 9x

D. no solution

Reflection: 18

Section 1.7 Solving Absolute Value Equations Objective(s): Solve absolute value equations in one variable.

Essential Question: Explain why are there two solutions to an absolute value equation.




Notes:

Vocabulary

The absolute value of a number is the distance between the number and 0 on the number line.

Ex: Solve the absolute value equation: 3x

Note: This means that the distance from the origin is 3.

Solution: There are two numbers that are 3 units away from the origin: 3 and 3x x

Solving an absolute value equation

When solving an absolute value equation in the form ax b c , you must solve the equations

ax b c and ax b c .

Ex: Solve the absolute value equation: 5 1 3 14x

Step One: Put the equation in ax b c form. 5 1 11x

Step Two: Rewrite as two equations. 5 1 11x 5 1 11x

Step Three: Solve both equations. 5 10x 5 12x

Solutions: 2x or 12

5x

Note: You can check your solutions by substituting them back into the original equation.

-2 -1 2 0 3 1 4 5 -3

Reflection: 19

What is the solution set of the absolute value equation?

Example 1: |x – 3| = 5

Equation 1: ___________________ Equation 2: ___________________

Solution 1: ___________________ Solution 2: ___________________

Example 2: |5x + 9| = 3


Example 3: |8x + 6| + 4 = 11


Example 4: |3x + 4| + 7 = 15



1. What is the solution set of 4 8 20x ?

A. { 7}

B. { 7,3}

C. {3}

D. { 3,7}

Documents

Section 1.1 Real Numbers and Number Operations - Edl · Section 1.1 Real Numbers and Number Operations ... Properties of Addition and Multiplication ... 2. Exponents (including square