advanced algebra honors. most problems are non-calculator …solving linear inequalities is the same...
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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:
Example 1: Example 2:
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
Example 3: Example 4:
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
8
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
9
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.
Example 1: Example 2:
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
Find the slope of the line connecting
(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
Find the slope of the line connecting
(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:
Example 1: Example 2:
Example 3: Example 4:
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
Example 1: Example 2:
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.
Example 1: Example 2:
#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.
3) Connect the points with a line.
Graphing a horizontal line or vertical line:
1) Determine two points on the line.
2) Connect the points with a 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
15
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
17
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
18
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
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