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Tutorial for the Algebra 2A/B Summer Assignment
****At the end of the tutorial, there are example problems to be
completed.
Ex. 1 Solve  6  3x  9  27
+9
+9
Add 9 to each section.
+9
3 3 x 36


3 3
3
Divide by 3.
1  x  12
Graph
< or > use an open circle when you graph
 or  use a closed circle
3
8 x  12  3x  53x  6  2 x  7 Distribute 3 to(8 x  12) and
4
4
 5to(3 x  6)
The result is 6x  9  3x  15x  30  2x  7
Combine Like Terms
Ex. 2 Solve
9x  9  13x  37
 13x
 13x
22x  9  37
+9
+9
22 x  28
Solve for x.
Divide by 22 and reduce if possible.
22 x  28

22
22
x
 14
11
Graph
Ex. 3 Solve and Graph
4x  7  9
For greater than absolute value inequalities you need to create two inequalities. First copy the original
problem without the absolute value symbol. The second inequality is created by reversing the inequality
and taking the opposite sign of the # on the right. Your inequalities will look as follows:
or
Solve the inequalities.
4x  7  9
4x  7  9
4x  7  9
+7 +7
or
4x  7  9
+7 +7
4x  16
4 x  2
4 x 16

4
4
4x  2

4
4
x4
or
x
1
2
Graph
This is an or problem because the inequality is either > or  , which means when we graph the solution
we take the union of the two graphs.
Ex. 4 Solve and graph 5x  6  9
or
+6 +6
3x  5  38
-5 -5
5 x 15

5
5
or
3 x 33

3
3
x3
or
x  11
Graph
Ex. 5 Solve and graph 3x  2  8
You also need to create two inequalities for a less than absolute value inequality, but you will write it as
one single expression. To do this you must begin by writing the original inequality without the absolute
value symbol, than place the opposite sign of the # to the right in front of the original inequality as shown
below.
 8  3x  2  8
-2
-2 -2
Follow Ex.1 to help you solve the inequality.
 10 3 x 6


3
3 3
Graph
Ex. 6 Solve and graph 4 x  5  13
Follow the steps from Ex. 3.
4x  5  13
-5 -5
or
4x  8
x2
4x  5  13
-5
-5
4 x  18
or
x
 18
reduce
4
x
9
2
Graph
Ex. 7 Solve. 3x  2  17
This is an absolute value equation. You create two equations by writing the original equation without the
absolute value symbol, and the second equation is the same except you take the opposite sign of the
number on the right. See below.
3x  2  17
-2
-2
or
3x  2  17
-2
-2
3x  15
3x  19
3 x 15

3
3
3x  19

3
3
x
x5
Ex. 8 Solve. 7  3x  28
7  3x  28
-7
-7
 19
3
Follow the steps from example 7.
or
7  3x  28
-7
-7
3x  21
3x  35
3 x 21

3
3
3 x  35

3
3
x7
x
 35
3
For examples 9 – 11, write the equation of the line, then graph the line.
Ex. 9 Given two points (5,3) and (-2,10), find the equation of a line.
First find the slope given by the formula m 
y 2  y1
10  3
7
 1
. m
=
25 7
x2  x1
Next use the slope-intercept form of an equation which is y  mx  b where m = the slope and b = the
y-intercept. Substitute in for x and y using any given point and substitute in for m from step one to find b.
y  mx  b
3  1(5)  b
3  5  b
8b
Now substitute in for m and b into the equation y  mx  b.
Answer: y   x  b
Ex. 10 x-intercept is -4
This is the point (-4,0).
This is the point (0,7).
y-intercept is 7
Now find the slope and follow the steps in example 9.
m
70
7

0  (4) 4
y  mx  b
I chose to use point (0,7)
7
( 0)  b
4
b7
7
Answer:
y
7
x7
4
Ex. 11 f (4)  3,
y
x
Point 1 (4,-3)
f (2)  5
x y
Point 2 (2,5)
Now find the slope and follow the steps in example 9.
m
5  (3)
8

 4
24
2
y  mx  b
5  4 ( 2 )  b
5  8  b
13  b
Answer:
y  4 x  13
Ex. 12 You want to fence in your yard. The area that you want to enclose is 800 square feet. The
length of the property is 40 feet. How much will it cost you to put a fence around your property if fencing
costs $35 a foot?
Area = length  width
A  lw
800  40 w
800 40 w

40
40
20  w
40
20
20
40
Now find the perimeter:
P  2l  2w
P  2(40)  2(20)
P  80  40  120 feet
The cost for fencing will be 120 feet  $35 = $4,200.
Ex. 13 A plumber charges $65 to enter your home and $45 per hour of work.
a) Write an inequality that represents the possible number of hours the plumber could work for $335.
$65 represents your fixed cost, $45x represents the number of hours worked times $45, and $335
represents the maximum cost, thus the following equation:
$65  45x  $335
b) Solve the inequality: 65  45x  335
-65
-65
45 x  270
45 x 270

45
45
Answer: x  6
Ex. 14 The length of a soccer field can vary from 80 feet to 120 feet inclusive. Write an inequality that
describes the possible lengths of a soccer field.
Answer: 80  x  120
Ex. 15 On your first five tests of the marking period, you earn grades of 95, 90, 80, 88, and 85. What
grade would you need to earn on your last test so that you have an average of 88.
Since you will have taken six tests and the average of the six tests is an 88, multiply 6  88. This will
give you the total points needed for an 88 average after 6 tests.
Total points will equal 528 (6  88=528).
Now take the sum of the five tests and subtract that total from 528.
95+90+80+88+85=438
Total Points:
Minus the sum of the 5 scores:
528
-438
Score needed to average an 88:
90
Answer: You need a 90 on your last test to get an 88 average.
Ex. 16 Graph the function
f ( x)  mx  b
m = the slope =
f ( x) 
is the same as
rise
run
2
x 5.
3
y  mx  b
b = the y-intercept, which is the 1st point you plot on the y-axis.
Step 1: Plot -5 on the y-axis.
Step 2: From -5, the slope tells you to go up two and right 3 where you will plot your 2nd point. Connect
2
the two points and you now have the graph of f ( x)  x  5
3
Ex.17 Graph the following inequality: y  5 x  3
m  the slope =
rise  5

run
1
b  y  int ercept  3
Follow Ex. 16 for directions on how to graph y  5 x  3.
Shade below the line because it is less than or equal to.  or  is a solid line because the points on the
line are included. < or > is a dashed line because the points on the line are not included.
Ex.18 Here is an example of a vertical line test.
a.
b.
This is an example of a graph that passes the
vertical line test. When you draw vertical lines
they intersect at only one point.
This graph fails the vertical line test.
When you draw vertical lines they intersect at
more than one point.
Ex. 19 Graph the following absolute value function: y  2 x  2  3 . This is in the form of
rise
. To graph the function, start at point (0,0). H moves the point (0,0)
run
left or right. For ex., x  2 moves the graph 2 units to the right.
y  a x  h  k . a = the slope =
x  2 moves the point 2 units to the left. Wherever h lands on the x-axis, move up or down according to
what k is. 2 moves the point up 2 units and -2 down 2 units
y  2 x  2  3. From (0,0) move left 2, then down 3 and plot your first point at (-2,-3).
rise
a = the slope =
. From the point (-2,-3) the slope tells you to rise 2 then plot 2 additional points left
run
1 and right 1 to form a v figure. See the graph on the following page.
Ex. 20 Write the absolute value function that has the following properties:
a) Reflects over the x-axis: this means the graph opens downward.
b) Has a vertical stretch of 3: this means the slope equals a =
3
.
1
c) Has a horizontal shift of 4: this means 4 the left from (0,0) and k = 4
d) Has a vertical shift up 4: from -4 on the x-axis move up 4 then plot the point (-4, 4)
Now substitute in the equation y  a x  h  k therefore y  3 x  (4)  4 . Now simplify to get
y  3 x  4  4 .
Ex. 21 Evaluate the following function for x = -2:
f ( x)  4 x 2  3 x  12
f (2)  4(2) 2  3(2)  12
f (2)  4(4)  6  12
Evaluate is another word for substituting in for x.
f (2)  16  6  12
f (2)  2
Follow the order of operations (PEMDAS).
Ex. 22 At 4AM, there was 3 inches of snow on the ground and at 2PM there were 23 inches of snow on
the ground.
a) Write a model for this situation.
First, find the slope using:
m = (the difference in snowfall)  (the difference in time).
m
23  3
20inches

 2inches / hour.
2 PM  4 AM 10hours
The linear model is y  2 x  3 ; 2x represents the snowfall/hour and 3 represents the original amount of
snow.
b) Based on your model, how much snow was there at 4PM?
From 4AM to 4PM is 12 hours and x = hours, therefore:
Ex. 23 Graph the line 5 x  3 y  15. Determine the x- and y- intercepts. Also, determine the slope of
the function.
To find the x-intercepts, set y = 0, then solve for x.
5 x  3(0)  15
5 x  15
Plot 3 on the x-axis.
x3
To find the y-intercept, set x = 0, then solve for y.
5(0)  3 y  15
 3 y  15
Plot -5 on the y-axis, then connect the points to get your graph of a line.
y  5
Now to find the slope of the equation put the equation into slope-intercept form ( y  mx  b ) by solving
for y.
5 x  3 y  15
The slope is m 
 3 y  5 x  15
y
5
x5
3
5
3
Ex. 24 Solve the following absolute value equation: y  4 x  7  12 for y  16.
16  4 x  7  12
28  4 x  7
7  x7
First substitute 16 in for y.
Then add 12 to both sides.
Then divide by 4.
Now, follow the steps from example 7.
x7  7
or
x  7  7
x = 14
or
x=0
Practice Problems
Solve, state the solution set, and graph. SHOW ALL WORK!!!!
1.  5  10 x  15  25
4
2. (20 x  45)  3x  4(7 x  4)  8 x  2
5
3. 3x  5  10
4. 4 x  3  9 or 7 x  1  43
5. 4 x  1  17
6. 3x  2  11
Solve the following absolute value equations
7. 7 x  5  9
8. 5  4 x  16
Write the equation of a line that passes through the given points. Then, graph the line.
9. (7, 2) and (-4, 6)
11. f (3)  2, f (8)  8
10.
x-intercept 5
y-intercept -6
SHOW ALL WORK!!!!!
12. You want to fence in your yard. The area that you want to enclose is 600 square
feet. The length of your property is 20 ft. How much will it cost you to put a fence
around your property if fencing costs $25 a foot?
13. A plumber charges $75 to enter your house and $40 per hour of work.
a. Write an inequality that represents the possible number of hours the plumber
could work for $515.
b. Solve the inequality.
14. The length of a standard hockey rink can vary from 80 ft. to 90 ft., inclusive. Write
an inequality that describes the possible lengths of a hockey rink.
15.On your first four tests of the marking period, you earn grades of 98, 93, 82, and 87.
What grade would you need to earn on your last test so that you have an average of
90?
16. In your own words, define what a function is. Then, graph the function:
f ( x) 
3
x 3.
4
17.Graph the following inequality: y  4 x  1 . Shade the appropriate region.
18.What is the vertical line test and how is it used? Write your answer using complete
sentences. Give an example of a graph that passes the vertical line test and an
example of a graph that fails the vertical line test.
19.Graph the following absolute value function: y  3 x  1  4
20.Write the absolute value function that has the following properties:
a.
b.
c.
d.
Reflects over the x-axis.
Has a vertical stretch of 7.
Has a horizontal shift of 9 to the right.
Has a vertical shift down 2.
21.Evaluate the following function for x = 3 : f ( x)  2 x  6 x  9
2
22. At 6AM, there was 6 inches of snow on the ground and at 1PM, there was 28 inches
of snow on the ground.
a. Write a model for this situation.
b. Based on your model, how much snow was there at 5 PM.
23.Graph the line 4 x  2 y  8 . Determine the x- and y-intercepts. Also, determine
the slope of the function.
24. Solve the following absolute value equation: y  3 x  5  6 for y = 9.