advanced algebra honors. most problems are non-calculator …solving linear inequalities is the same...

Algebra 2 Honors Summer Packet

1

Congratulations! You are going to be in Algebra 2 Honors!

Here is a packet of pre-algebra/algebra topics that you are expected to know before you start

Advanced Algebra Honors. Most problems are non-calculator. To prepare yourself for the year

ahead, do not use a calculator while doing these problems. If a calculator may be used to help with

the calculation process, it will be clearly marked: CALC.

This packet contains notes, worked out examples, and practice problems. Be sure to read the

examples in each section before starting. You can show your work directly in this packet.

This packet should be completed before the first day of school. We will go over the answers and

questions from this packet during first few days of school, so if you have a few questions, don’t

worry. Remember to bring your completed packet with you on the first day of school!

We look forward to meeting you in August!

Number Hierarchy:

Complex Numbers

Real Numbers

Rational Numbers

Integers

Whole Numbers

Natural Numbers

Irrational

Numbers

Imaginary Numbers

2

Definitions of Numbers:

Complex: A number a bi where a and b are real numbers and i is the imaginary number. All

numbers are complex.

Real Numbers: Numbers that can be graphed on a number line.

Imaginary Numbers: Numbers that contain the 1 i

Rational Numbers: Numbers that can be written as the ratio of two integers.

Irrational Numbers: Numbers that cannot be written as the ratio of two integers.

Integers: Positive and negative whole numbers.

Whole Numbers: The set of the natural numbers and zero.

Natural Numbers: The set of the counting numbers {1,2,3,4,5…}

1) Check the box if the value belongs to that given set.

Inte

ger

s

Even

Nu

mb

ers

Od

d N

um

ber

s

Posi

tive

Nu

mb

ers

Neg

ati

ve

Nu

mb

ers

Rati

on

al

Nu

mb

ers

Irra

tion

al

Nu

mb

ers

Imagin

ary

Nu

mb

ers

Rea

l N

um

ber

s

Natu

ral

Nu

mb

ers

Cou

nti

ng N

um

ber

s

Com

ple

x N

um

ber

s

a) 5

b) 2/3

c) -7

d) 3

e) 16

f) 16

g) 0

h) 44

i)

j) 1.765

k) -10000

l) 1.5

m) 6

3

Conversion of units

Example 1: Convert 11 yards to inches.

3 feet 12 inches11 yards 396 inches

1 yard 1 foot

Example 2: Convert 45mph to feet per second.

45 miles 1 hour 1 minute 5280 feet 66 feet per second

1 hour 60 minutes 60 seconds 1 mile

CALC: Evaluate by showing all steps!

1. Convert 350 feet to yards.

2. Convert 20 mi/hour to feet per second.

3. Convert 6 feet per second to mph.

4

7 10 9 6

10 2 6

4 2

2

p p

p

p

p

4 5 35q

6 15 10y 4(2 5) 8( 2)n n

48 25 12 11j j 4( 2) 4 8x x

Solving Linear Equations:

Example 1: Example 2:

38 29

5

321

5

5(21)

3

35

x

x

x

x

Solve the equation. Check your solution.

1. 5 30t 2.

3. 4.

Solve the equation. Check your solution.

5. 6.

5

2 34

3 5m m

1 2 14

2 3 2w w

2 1 16( )

3 12 8x x

Solving Linear Equations by Clearing Fractions:


1 110

2 3

1 1( 10)*62 3

3 2 60

5 60

12

x x

x x

x x

x

x

1 3 25( )

4 5 3

5 23

4 3

5 2( 3 )*124 3

15 36 8

15 28

28

15

x

x

x

x

x

x

Solve the following equations and check your answers.

1. 2.

Solve the following equations and check your answers.

3.

6

Rewriting Formulas

A formula is an equation that relates two or more variables.

To rewrite a formula is to solve for another variable and write the equation as an equivalent

equation.

Example 1:

Solve the distance formula for the rate.

d r t

dr

t

Example 2:

Recall the formula for the sum of all interior angles of a polygon, S = 180(n-2)

Solve this formula for n.

Method #1:

S = 180(n-2)

S = 180n – 360

S + 360 = 180n

( 360)

180

Sn

Method #2:

S= 180(n-2)

2180

2180

Sn

Sn

*Note: Both the formulas above are correct. They are mathematically equivalent.

Formulas you should be familiar with from previous courses:

Quantity Formula Meaning of variables

Distance d rt distance, rate, timed r t

Temperature 932

5F C

degrees in Fahrenheit

degrees in Celcius

F

C

Area of a triangle 1

2A bh

Area, base,

height

A b

h

Perimeter of a rectangle 2 2P l w perimeter, length,

width

P l

w

Area of a trapezoid 1 2

1( )

2A b b h 1 2Area, base 1, base 2,

height

A b b

h

Area of a circle 2A r Area, radiusA r

Circumference of a circle 2 or C r C d Area, radius,

diameter

A r

d

7

Rewrite the following formulas as requested.

1. Solve the temperature formula for degrees Celsius.

2. Solve the area of the trapezoid formula for one base.

3. Solve the slope-intercept equation y mx b for the slope, .m


Solve 7x – 4y = 5 for x. Solve 7x – 4xy = 5 for x

7x = 5 + 4y x(7 – 4y) = 5

(5 4 )

7

yx

5

7 4x

y

Solve the following equations for x.

1. y + 6x = 2 2. 8y – x = 9

Solve the following equations for x.

3. 4x – 2xy = 7 4. 2xy – x = 3

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Graphing Linear Inequalities

The graph of an inequality in one variable consists of all points on a number line that represent

the solutions. Recall that an “open” circle means the value is not included in the solution and a

“closed” circle means the value is included in the solution.

Examples:

4x 2x

<----------------------------------------> <---------------------------------------->

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

3 2x <--------------------------------------------->

(x > -3 and x < 2) -4 -3 -2 -1 0 1 2 3 4

2x or 1x <--------------------------------------------->

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

Important note: “and” means that all values in the solution make BOTH conditions true. This is

the “overlap” from the two given condition. “or” means that all values in the solution make at

least ONE condition true. It is possible to have a “no real numbers” or an “all real numbers”

case.

2x and 1x has no real solution (no “overlap”)

2x or 1x all real numbers will make this condition true!

Graph each inequality on the number line & simplify inequality if possible.

1. 3x and 2x <--------------------------------------------->

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

2. 1x or 2x <--------------------------------------------->

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

3. 2x or 1x <--------------------------------------------->

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

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Solving Linear Inequalities

A linear inequality in one variable can be written in one of the following forms, where a and b

are real numbers and 0a .

0 0 0 0ax b ax b ax b ax b

Solving linear inequalities is the same as solving linear equations with one exception -- if you

multiply or divide BOTH sides of an inequality by a NEGATIVE number, then you must

reverse, or “flip,” the inequality sign.

A solution of an inequality in one variable is a value that, when substituted for the variable,

results in a true statement.

Examples: Solve the inequality. Also, check your inequality by testing values for x that are

inside and outside your solution set.


2 4 8x x

3 2 8x

3 6x

2x

Check 2 4 8x x

If x = 2, 2-2 = 4(2) – 8 true

If x = 3, 3 – 2 < 4(3) – 8 true

If x = 0, 0 – 2 < 4(0) – 8 false, so x = 0

should not be in the solution (and it isn’t)

4 5 9 10x x

5 5 10x

5 15x

3x

Check 4 5 9 10x x

If x = 4, 4(4) + 5 >9(4)-10 false

If x = 3, 4(3) + 5 > 9(3) – 10 false

If x = 0, 4(0) + 5 > 9(0)-10 true

Example 3:

7 5 2 8x

5 5 10x

1 2x

(this means x > -1 and x< 2)

Check 7 5 2 8x

If x = 0, -7 <5(0) – 2 < 8 true

If x = -1, -7 <5(-1) – 2 < 8 false

If x = 2, -7 <5(-2)-2< 8 true

Example 4:

4 7 5x or 3 2 23x

4 12x or 3 21x

3x or 7x

Check 4 7 5x or 3 2 23x Recall that each value for x just needs to make

one condition true (because of the “or”)

If x = 0, 4(0) – 7 < 5 true

If x = 10, 3(10) + 2 > 23 true

10

2 1

2 1

y yy risem

x x x run

Find the slope of the line connecting

(3,5) and (7,2)

2 5 3

7 3 4

Oblique Line!

ym

x


(2,5) and (2,-8)

8 5 13

2 2 0

Vertical Line!

ym undefined

x

Find the value of k given the slope of the line

1(2,4) and (3k, k+1) with slope=

6

1 4 1

3 2 6

3 1

3 2 6

6( 3) 1(3 2)

6 18 3 2

3 16

16

3

y k

x k

k

k

k k

k k

k

k


(2,4) and (7,4)

4 4 00

7 2 5

Horizontal Line!

ym

x

Solve the inequality. Also, do a “mental” check your inequality by testing values for x.

1. 3 5 8 25x x 2. 14 9 20 33x x

3. 5 3 2 7x 4. 3 1 10x or 3 4 11x

Finding the Slope of a Line/Rate of change

Formula:



11

3 4

3(0) 4

4

y x

y

y

For the following, find the slope and identify if the line is vertical, horizontal or oblique.

1. (4, 4) and (4, 8) 2. (2, -2) and (8, -2)

3. (0, 8) and (9, 0)

4. Find the value of k so that the line through the given points has the given slope.

(2, -3) and (k, 7); m=-2

X and Y Intercepts


The x-intercept(s) occur when y = 0. The y-intercept occurs when x = 0.

To find the x-intercept, set the y-value To find the y-intercept, set the x-value

equal to zero and solve. equal to zero and solve.

2 24

0 2 24

24 2

12

y x

x

x

x

For #1-6, a) Find the x- intercept b) Identify the y- intercept and c) Identify the slope

1. 3

32

y x 2. 2 4y x

3. 4 2 8x y 4. 5 2 10x y

12

Write the equation of the line in all forms

connecting: (-2,5) and (3,1)

1 5 4

3 ( 2) 5

Point-Slope Form:

4 41 3 or 5 2

5 5

Slope-Intercept Form:

4 17

5 5

Standard Form:

4 5 17

ym

x

y x y x

y x

x y

Given:

1 and (2,1)

3

Point-Slope Form:

11 2

3

Slope-Intercept Form:

1 1

3 3

Standard Form:

3 1

m

y x

y x

x y

(4, 7) &(0,9) ( 2,3) &(4,7)

Equations of Linear Functions

Point – Slope Form of a line is: 1 1( )y y m x x where m is the slope of the line and

1 1( , )x y is a point on the line.

Slope – intercept form of a line is: y mx b where m and b are constant values. The slope

of the line is m and b is the y-intercept (0, b ).

Standard Form of a line is: Ax By C where , ,&A B C are integer values and A is positive.


#1-3. Write each line in all three forms, if possible. If not possible, write in correct form.

1. Line connecting: 2. Line connecting:

3. Line with slope: -3 and goes through (7,2)

13

26

3y x

Graphing a Line in any form

Given a linear function, use the following methods to graph the line.

Graphing a Line in any form:

1) Construct a table of values.

2) Plot enough points from the table to recognize a pattern.

3) Connect the points with a line.

Graphing a Line in Point-Slope Form:

1) Determine the slope and point of the line.

2) Graph the given point.

3) Use the slope (m) and use it to plot at least a second point on the line.

4) Draw a line through the two points.

Graphing a Line in Slope-Intercept Form:

1) Determine the slope and y-intercept of the line.

2) Plot the y-intercept coordinate (0,b).

3) Use the slope (m) and use it to plot at least a second point on the line.

4) Draw a line through the two points.

Graphing a line in Standard Form:

1) Determine the x-intercept by plugging zero in for y and solving for x. Plot the point.

2) Determine the y-intercept by plugging zero in for x and solving for y. Plot the point.


Graphing a horizontal line or vertical line:

1) Determine two points on the line.


Example 1: Graph

m = 2

3(up 2, right 3)

b = 6 (0,6)

14

:5 2 10F y x

: 1 2( 3)F y x

Graph lines A-D on the same coordinate grid. Label each line.

1: 3

3A y x

: 2 4 12B x y

: y = 4C

: 3D x

Graph lines E-H on the same coordinate grid. Label each line.

: y =E x

1: 3 ( 2)

2G y x

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Systems of Equations

Systems of equations are classified as consistent or inconsistent.

A consistent system has at least one solution. If a system of equations is classified as

consistent, it can be either independent or dependent.

If the consistent system is independent, then it has one solution.

If the consistent system is dependent, then it has infinitely many solutions.

An inconsistent system has no solutions.

Methods for Solving a System of Equations. Solve each system by using the indicated method.

Then, classify the system.

Examples:

Graphing

4 2 4

2 3 10

x y

x y

The solution is 2, 2 and it is a

consistent, independent system.

Substitution

8 2 2

3 14

x y

x y

1. Solve equation (2) for y:

14 3x y

2. Substitute into equation

(1):

8 14 3 2 2y y

112 24 2 2y y

112 22 2y

22 110y

5y

3. Substitute value of y into

either equation to solve

for x:

14 3 5x

1x

The solution is 1,5 and it

is a consistent, independent

system.

Elimination

4 2 2

6 5

x y

x y

1. Multiply equation (2) by

2:

4 2 2

12 2 10

x y

x y

2. Add equations (1) and

(2):

16 8x

3. Solve for x:

1

2x

4. Substitute value of x into

either equation to solve

for y:

16 5

2y

2y

The solution is 1

,22

and

it is a consistent,

independent system.

x

y

16

x

y

x

y

1. Solve the system by graphing and then classify the system.

a.

3 4 8

32 6

2

x y

x y

b. 3 2

5 2 2

y x

x y

2. Use any method that you would like to solve the system of equations. Then, classify the

system.

a. 3 2 1

4 6 7

x y

x y

b.

2 3 3

4 6 8

x y

x y

c.

2 5 3

4 10 6

x y

x y

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Use Problem Solving Strategies & Models

When solving “real world” problems, it is helpful to write an equation in words before you write

the equation in mathematical symbols. This word equation is called a verbal model. Verbal

models can be formulas, patterns, or diagrams. Here are some examples:

Example 1: Use a formula.

A bus travels at an average rate of 55 miles per hour. The distance between Chicago and San

Francisco is 2130 miles. How long would it take for the bus to travel from Chicago to San

Francisco?

Verbal model Distance = Rate * Time

& Formula D=rt

2130 = 55t

T = 2130

55 hours or approximately 36.9 hours

Example 2: Look for a pattern.

The table below shows the height h of a jet airplane t minutes after beginning its decent. Find

the height of the airplane after 15 minutes.

Time (min), t 0 2 4 6 8

Height (ft), h 35,000 32,000 29,000 26,000 23,000

Pattern Observed: The height decreases by 3,000 feet every two minutes.

Verbal Model Height = initial height – (rate of descent)*(time)

& Formula H = 35000 – (3000/2)t

H = 35000 – 1500t

H = 35000 – 1500(15)=12,500 feet after 15 minutes

Example 3: Draw a Diagram.

You want to paint five 1-foot wide stripes on the wall. There should be an equal amount of

space between the ends of the wall and the stripes and between each pair of stripes. The wall is

14 feet long. How far apart should the stripes be?

Diagram:

Equation to solve: x + 1 + x + 1 + x+ 1 + x + 1 + x+ 1 + x = 14

6x + 5 = 14 6x = 9 x = 1.5 feet between each stripe

x x x x x x

1 ft 1 ft 1 ft 1 ft 1 ft

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CALC: Use one of the problem solving strategies to answer each question. Show all your

work, as demonstrated by the examples on the previous page.

1. If a jet airplane descends at the rate given in the table, what is its height after 27 minutes?

Time (min), t 0 4 8 12 16

Height (ft), h 46,000 42,800 39,600 36,400 33,200

2. A car used 16 gallons of gasoline and traveled a total distance of 460 miles. The car’s fuel

efficiency is 30 miles per gallon on the highway and 25 miles per gallon in the city. How

many gallons were used on the highway? (The verbal model is provided for you this time

Total distance

(miles) =

Highway

fuel efficiency *

Gas used

on highway +

City fuel

efficiency *

Gas used

in city

3. A moving company weighs 22 boxes you have packed that contain either books or clothes.

The total weight of these boxes is 445 pounds. If each box of books weighs 35 pounds and

each box of clothes weighs 10 pounds, how many box of books did you pack?

4. If the perimeter of a rectangle is 120 meters, and the length is 40 meters, find the width of the

rectangle.

19

5. Your local cable company charges $30 per month for basic cable. Premium channels are

available for a surcharge of $6 per channel. You have $70 a month budgeted for cable. How

many premium channels can you purchase?

6. You want to hang six 2-foot wide posters on the wall. There should be an equal space between

the posters and you also want the spaces to the far right and far left of the poster group to be

twice the space between any two adjacent posters. The wall is 54 feet long. How far apart

should the posters be? (Express your answer in feet & inches.)

7. You want to hang six 2-foot wide posters on a cylindrical kiosk that has a diameter of 10

feet. There should be an equal space between the posters. How far apart should the posters

be? You may round to the nearest tenth of a foot.(Hint #1: Draw a diagram with an “unrolled” kiosk

…. Recall from geometry that the circumference of the kiosk is equal to the length of the unrolled rectangle.

Hint #2: Because the posters are hanging on a cylinder-shaped wall, think about how many spaces there are

between the posters.)

20

8. On a track at an Air Force base in New Mexico, a rocket sled travels 3 miles in 6 seconds.

What is the average speed in miles per hour.

9. At a vegetable stand, you bought 3 pounds of peppers for $4.50. Green peppers cost $1 per

pound and orange peppers cost $4 per pound. How many pounds of each kind of pepper did

you buy?

10. You have two summer jobs. In the first job, you work 25 hours per week and earn $7.75 per

hour. In the second job, you earn $6.25 per hour and can work as many hours as you want.

You want to earn $250 per week. How many hours must you work at the second job?

11. A quarter mile running track is shaped as shown. The formula for the inside perimeter is:

2 2P r x . Solve the perimeter for r.

12. Over a 30 day period, the amount of propane in a tank that stores propane for heating a home

decreases from 400 gallons to 214 gallons. What is the average rate of change in the amount

of propane? (Include units)

x r r

advanced algebra honors. most problems are non-calculator …solving linear inequalities is the same...

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