Description
In This collection, we will go deep into math.
This collection will help all math and high school students.
Study Set Content:
Luttrell 2012
61
Name: ______________________
Date: _____
4e - Evaluating
Most times you need to determine the value of an expression (the number an expression
represents). In order to find the value, you need to know what the variables represent. By
substituting the value of the variables and simplifying, the value of the expression will be found.
Note the vertical alignment of the work; try to show your work in similar manner.
Example A: Evaluate the expression when
y
= 2:
3
y
+ 2
y
2
- 6(
y
- 4) + 8
y
3
.
Step I - substitute value of
y
. 3(2) + 2(2)
2
- 6(2 - 4) + 8(2)
3
.
Step II - simplify. 6 + 2(4) - 6(-2) + 8(8)
6 + 8 + 12 + 64
90
Example B: Evaluate
3 2
1
2
3
6
(
)
(
)
x
x
x
+ −
−
+
when
x
= -3
3 2
3
1
2
3
3
3
6
( (
)
)
(
)
− + − − −
− +
3
6
1
2
6
3
(
)
(
)
− + − −
3
5
12
3
(
)
− +
−
= −
3
3
1
Evaluate when a = -1.
1. (4a + 3)a - (2 + 2a)
2. 12a
2
- 3a + 2
3. 3a
4
+ 5a
3
- 6a
2
- 2a
Evaluate when
x
= 1,
y
= 2, and
w
= 2
4.
2
3
5
3
x
x
+ + −
5. -(-
x
)
6.
x
- (
x
- {
x - y
})
7. 4
w
+ 7 + 3
w
-2 + 2
w
8.
1
3
3
1
w
w
w
w
w
w
w
+ × +
× −
9.
3
2
5
(
)
(
)
x
y
x
y
x
y
+
−
−
+
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Name: ______________________
Date: _____
4f- Expanded Form
Do without the aid of a calculator. Check your answers with a calculator at the end.
Fill in the missing parts by writing out the numeric value.
1. 8,375 = eight thousand, three hundred __________________
2. 312, 435, 600 = three hundred twelve million, ___________________________, six hundred
3. 412 = ____________________________ twelve
4. 8,300,567 = eight million, ____________________________, five hundred sixty-seven
Identify the numeric value of the number indicated:
Example 567,890
5 represents the _______hundred thousands______
5. 320,045
2 represents the ____________________________
6. 1030
3 represents the ____________________________
7. 321
3 represents the ____________________________
Write out in expanded form:
Example: 431.24
4(100) + 3(10) + 1(1) + 2(1/10) + 4(1/100)
8. 3, 243
9. 345,645
10. -23,120
11. 304.125
12. 300,561, 100
Write out in standard form:
Example: 3(1000) + 4(10) + 8(1) + 9(1/10) = 3,048.9
13. 4(1,000,000)+8(10,000)+3(1,000)+7(100) =
14. 2(10,000) + 8(100)+3(10)+4(1/10) =
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Name: ______________________
Date: _____
4g-Expanding Numbers Using Powers
Use calculator only when checking answers!
Write the following as a power of 10:
1. 10,000
2. 100
3. 10,000,000
4. 1
Write each of the following in standard form:
5. (4 ×10
4
) + (9 ×10
3
) + (5 ×10
2
) + (6 ×1)
6. (6× 10
5
) + (6×10
2
)
7. (7×10
6
) + (8×10
5
) + (6×10
4
)
8. (8×10
4
) + (2× 10
3
) + (3 × 10
2
)
9. (4×10
2
) + (8×10
4
) + (4×1) + (7×10
3
)
Write the following numbers in expanded form, using powers:
10. 34,500
11. 403
12. 1,234
13. 50,500,000
14. 271,828,459,045
Luttrell 2012
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Name: ______________________
Date: _____
4h- Expanded Form using Powers
Do without the aid of a calculator. Check your answers with a calculator at the end.
Use the powers of 10 to expand this number:
example: 8,321 = 8(10
3
) + 3(10
2
) + 2(10
1
) + 1(10
0
)
1. 340, 073
2. 53,100
3. 830,401
4. 234,876,912
5. 32
6. 144
7. 300,045
8. 16, 230, 030
Write the following numbers in standard form:
9. 7(10
3
) + 4(10
2
) + 2(10
1
) + 6(10
0
)
10. 8(10
5
) + 8(10
3
) + 4(10
1
) + 7(10
0
)
11. 8(10
6
) + 1(10
5
) + 2(10
4
) + 1(10
3
) + 5(10
2
) + 3(10
1
)
12. 5(10
8
) + 8(10
6
) + 7(10
4
) + 9(10
2
) + 2(10
0
)
We work with base 10, probably because we have 10 fingers. But other societies used different
bases. France used to have base 20 (they used fingers and toes). It still exists in some names for
numbers, like
quatre-vingt-neuf
. The Babylonians used base 60 which remains in our time: 60
minutes in an hour and 60 seconds to a minute. Computers made wide use of counting by 2s or
16s. Follow the example of how to convert other bases into base 10 (standard form).
Example: 11101
2
= 1(2)
4
+ 1(2)
3
+ 1(2)
2
+ 0(2)
1
+ 1(2)
0
= 16 + 8 + 4 + 0 + 1= 29
10. 10111
2
= 1(2)
4
+ 0(2)
3
+ 1(2)
2
+ 1(2)
1
+ 1(2)
0
= _________________________________
11. 101
2
= 1(2)
2
+ 0(2)
1
+ 1(2)
0
= _________________________________
12. 110101
2
= 1(2)
5
+1(2)
4
+ 0(2)
3
+ 1(2)
2
+ 0(2)
1
+ 1(2)
0
= _____________________________
13. FACE
16
= F(16)
3
+ A(16)
2
+ C(16)
1
+ E(16)
0
= 15(16)
3
+ 10(16)
2
+ 12(16)
1
+ 14(16)
0
= ____________________________________________________ (calculator allowed)
Luttrell 2012
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Name: ______________________
Date: _____
4i-More Practice with Standard and Non-Standard Forms
Exercise A. Write the following numbers in expanded form:
Example: 835 = 8(10
2
) + 3(10
1
) + 5(10
0
)
1. 435,800
2. 12,300,506
3. 1000
4. 15,000
5. 4,000,000,000,000,000
6. 5,000,000,000,000
Exercise B. Write the following numbers in expanded form, using the example as a guide:
Example: 1.2345 =
1 2
3
4
5
1
10
1
100
1
1000
1
10 000
+
+
+
+
( )
(
)
(
)
(
)
,
7. 12.005
8. 0.0004
9. 0.100304
10. 4.500005
Exercise C. Write the following numbers in expanded form, using the example as a guide:
Example: 1.2345 =
1 2 10
3 10
4 10
5 10
1
2
3
4
+
+
+
+
−
−
−
−
(
)
(
)
(
)
(
)
11. 12.005
12. 0.0004
13. 0.100304
14. 4.500005
Exercise D. Write the following numbers in standard form.
15. 8(10
6
) + 3(10
5
) + 5(10
4
) + 1(10
0
) =
16. 4(10
13
) + 8(10
10
) + 5(10
5
) =
17.
3 10
5 10
4 10
5 10
3
0
1
2
(
)
(
)
(
)
(
)
+
+
+
−
−
=
Luttrell 2012
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Name: ______________________
Date: _____
4j - Scientific Notation
Since mathematicians and scientists have tried to find ways of simplifying written expressions, scientific
notation is common for writing VERY large or small numbers. This is very useful when talking about the
distance between stars, light years, the number of molecules in an object, the people on the planet, etc.
Note how this large number is rewritten:
64000000 = 6.4∙10
7
The 6.4 is the mantissa and is always a
number between 1 and 9, inclusively. The exponent 7 is the characteristic and says what power of 10 you
are multiplying by. Some people just remember that is how many spaces to move the decimal point. A
negative moves it left and a positive moves the decimal right. A small decimal like 0.000007 is rewritten
as
0.000007 = 7∙10
-6
.
1. 0.000123
2. 2,340,000,000
3. 0.0000000345
4. 435000000000000
5. 76,000,000,000
6. 0.0000308
Express each number into scientific notation. For some you may have to round to three
significant digits in the mantissa.
7. 23,000,000,000,000
7. ______________________
8. 134,000,000,000
8. ______________________
9. 456,000,340,000,000
9. ______________________
10. 788,999,213,543,345,532,345
10. ______________________
Express each number in standard form.
11.
2 34 10
4
.
×
11. ______________________
12.
3 4 10
14
.
×
12. ______________________
13.
6 7 10
7
.
×
13. _______________________
14.
2 6 10
9
.
×
14. _______________________
15. 3.4 × 10
-5
15. _______________________
16. 1 × 10
-10
16. ______________________
17. 3.25 × 10
-6
17. ______________________
Fill in the missing spots of the equation.
18.
0 0000234
2 34 10
.
.
=
×
19.
31 10
00000031
5
.
×
=
−
20
. 0.0045 = 45∙10
-3
21
. 6.1 ∙ 10 = 610,000
Luttrell 2012
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Name: ______________________
Date: _____
4k - Percents
Write the equivalent percent for each of the following:
1. 0.04
2. 0.12
3. 0.45
4. 1.23
5. 1
Convert the percent into standard decimal form:
6. 1%
7. 23%
8. 123%
9. 0.3%
10. 56%
Solve using percents:
11. What is 10% of $32?
12. What is 5% of $32?
13. What is 15% of $32?
14. What is 5% of $42?
15. What is 10% of $42?
16. What is 20% of $42?
17. What is 15% of $42?
18. What is 8% of $40?
19. What is 110% of 80?
20. A suit at JcPenny’s has a $185 tag. But it’s on sale for 30% off. How much will it cost without tax?
If the Nova Scotia government has a provincial tax of 15%, how much will the suit cost?
21. If you spent $42.75 on 3 CDs, how much does each one cost?
What percent of the total cost is one CD?
22. A dress has a price tag of $90. The dress is on sale for 20% off and there is a 6% sales tax.
What is the total cost of the dress? How much money do you save?
23. A dress has a price tag of $86.
a. If it is on sale for 20% off, what will be the cost? (Ignore tax)
b. What is the price of the dress on sale with a 6% tax?
c. How much would the dress cost when not on sale?
Luttrell 2012
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Name: ______________________
Date: _____
4L – Using Proportions with Conversions
Show your work! Calculator is allowed.
Going between Canada and the United States, you will find that the two countries use different
measuring systems. If you want to know how fast to go or how far away some city is, you might
have to do a bit of converting. Here are some basic units:
1 mile = 1.6 kilometer
10 mm = 1 cm
10 cm = 1 dm
10 dm = 100 cm = 1 m
1000 m = 1 km
2.5 cm = 1 in
12 in = 1 ft
3 ft = 1 yd
5280 ft = 1 mi (mile)
Metric Distance Conversion
1. 5 m = _____ cm = _____ mm = ______ km
2. 60 km = ______ m = _____ cm = ______ dm
3. 45 mm = ______ m = _______ km
4. 0.23 km = _____ m = ______ mm
5. 0.003 km = ______ m = ______ cm
Area Unit Conversion
1. 10 feet squared = _______ inches squared
2. 196 in
2
= ______ ft
2
3. 105 in
2
= ______ yd
2
4. 144 mm
2
= _____ cm
2
5. 10000 cm
2
= ______ m
2
6. 100 m
2
= ______ km
2
7. 145 m
2
= _____ km
2
8. 5 ½ m
2
= _____ cm
2
9. 10 ¾ yd
2
= ______ ft
2
10. 12 ¼ ft
2
= ______ in
2
11. 400 ½ m
2
= ______ km
2
12. 2 ¼ ft
2
= _____ in
2
13. 3 1/3 yd
2
= _____ in
2
14. 2 ¾ km
2
= ______ cm
2
Luttrell 2012
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Name: ______________________
Date: _____
4m - Proportions
Show your work! Use a proportion to solve each problem. Calculator is allowed.
1.
3
5
15
=
x
2.
x
x
+
−
=
1
5
1
2
3.
x
x
+
=
8
2
3
4. The ratio of seniors to juniors is 2:3. If there are 21 juniors, how many seniors are there?
5. A 15-foot building casts a 9-foot shadow. How tall is a building that casts a 30-foot shadow
at the same time?
6. A photo that is 3 inches wide by 5 inches high was enlarged so that it is 12 inches wide. How
high is the enlargement?
7. Philip has been eating 2 hamburgers every 5 days. At that rate, how many hamburgers will he
eat in 30 days?
8. An architect wants to build a model of the structure he is making. The structure is 80 feet tall
and 35 feet wide. His model will be 50 cm wide. How tall will it be?
9. Draw a rectangle whose dimensions are 3 cm by 4 cm. Find its area and perimeter. Then
draw an enlarged rectangle whose dimensions are 9 cm by 12 cm. Find its area and perimeter.
How do those numbers relate to the scale factor between the two images?
The following is a review of shapes. You may use a calculator.
10. A right triangle has two sides of 5 and 12. What is the distance of the hypotenuse?
11. What is a 4-sided polygon called?
12. What is a 6-sided polygon called?
13. If two angles in a triangle are 30 and 45, what is the third angle?
14. What is a scalene triangle, isosceles triangle, and equilateral triangle?
What relationship is there between the angles and sides being equal?
15. Draw an example of a reflection, rotation, and translation. Label each drawing.
Luttrell 2012
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Name: ______________________
Date: _____
4n – Mixed Review
Solve the following. Show your work! Calculator is allowed.
1
. If you drive 650 miles in a car uses 30 gallons, what is your average mpg?
2. If a house is 2.3 km away, how far is it in meters? Millimeters?
3. A teacher wants to show a movie in her 50 minute class. If the movie is 210 minutes long,
how many classes must she need in order to watch the entire movie?
4. A lawn of 100 ft by 150 ft has a house built on it. The house is 50 ft by 30 ft.
What percent of the lawn is left to landscape?
5. Amy used to work 60 hours each week. Now she works 45 hours a week.
What is the percent decrease?
Luttrell 2012
71
Chapter 4 Test
Name: ___________________________
Date: _____________
SHOW WORK
for full credit!!! You must work alone. Only a calculator is allowed.
1. Determine which ratio is larger. Use the symbols <, >, or = to fill the blank:
a.
5
6
____
8
9
b.
11
15
____
23
27
c.
15
18
____
5
6
2. Write the ratios:
a. There are fourteen P-3 students, fourteen 4-6th grade students, six 7-8th graders,
fourteen10
th
graders and nine 11-12th grade students at SLA. Write the ratio of the
4-6th to the 7-8th graders.
b. Ethan and Ben played checkers. Ethan won 8 games and Ben won 12. Write the
ratio, in simplest form that compares Ethan’s score to Ben’s.
3. Write each rate in simplest form, and give its units:
a. Kelsey skied downhill twice in 30 minutes.
b. The Morash family visits Martock 20 times in five months.
C. My granny bought 4 tickets to the Tattoo for $160.
4. Solve:
a.
4
24
54
x
=
b.
40
8
5
n
=
C.
12
13
5
=
x
5. Solve these word problems:
a. The ratio of the boy’s shadow to the flag pole’s shadow is 1:10. If the boy is 160 cm tall,
how high is the pole?
b. On a map 2.5 cm represents 300 km. What distance would a 3.5 cm line represent?
6. A. Which is cheaper: $3.20 for 8 L or $2.40 for 4 L?
b. Paul’s family travelled 190 km in 3 hours. At this speed, how far do they go in 5 hours?
5
5
5
5
5
5
Luttrell 2012
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Chapter 4 Test, continued
7. Answer each part:
a. What is 40% of 70?
b. What is 25% of 40?
c. Fifteen is 10% of what number?
d. Sixteen is 25% of what number?
e. Write as a percentage: 7 girls in a class of 18 students.
8. A city recycles 78% of the newspapers sold there. The Chronicle Herald has a readership of
316,700. How many of the Chronicle Herald gets recycled?
9. It costs you $25 to make a sweater and want to sell it for profit at $45. What percent markup
do you have (percent increase)?
10. SLA has 73 students and is praying for 100 next year. What is the percent increase?
11. Answer the following short questions:
a. Write as standard form: 2.34×10
5
.
b. Write as a sum of powers: 300.04.
c. Write as standard form: 6(10
3
) + 3(10
2
) + 4(10
0
)
d. Simplify: 3-3×2
3
∕4 – 3
e. Evaluate 3x
2
– 4x + 1 when x = -1.
Bonus: (5pts) Two identical jars are filled with equal number of marbles. The marbles are
colored red or white. The ratio of red to white in jar I is 7:1 and 9:1 in jar II. If there are 90
white marbles all together, how many red marbles are in Jar II?
10
5
5
5
5
Luttrell 2012
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Chapter 5 Number Line and Cartesian Plane
This chapter contains worksheets on the topics of solving linear equations and inequalities,
absolute value equations and inequalities, graphing on number lines, and plotting on the
Cartesian coordinate system.
Prior Skills
•
For sheet 5h, students need to understand what constitutes a polynomial.
•
For sheet 5i, domain and range need to be introduced.
Luttrell 2012
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Name: ______________________
Date: _____
5a – Solving for a Variable (Multiple Variables present)
When solving for a variable, use the inverse functions. To remove a division, multiply. To
remove an addition, subtract. Here is a list of some common inverse pairs:
add
multiply
square,
cubes...
logarithms
f(x)s
trigonometr
ic f(x)s
derivatives
subtract
divide
square root,
cube root...
exponential
f(x)s
inverse trig integrals
When faced with multiple variables, just focus on the variable you need to solve for. Remove all
the terms and coefficients that surround the needed variable. Here is an example:
Solve for
b
1
:
A
b
b h
=
+
1
2
1
2
(
)
2
1
2
A
b
b h
=
+
(
)
Since
b
1
is interior, remove constants
outside
2
1
2
A
h
b
b
=
+
of the parentheses first. Undo division of 2
2
2
1
A
h
b
b
−
=
by multiplying by 2. Undo multiplication of
h
by dividing. And so forth.
1. Solve for
h
:
V = ⅓πr
2
h
2. Solve for
m
:
y = mx + b
3. Solve for
b
:
V
b h
=
1
3
2
4. Solve for
r
:
V = πr
2
h
5. Solve for
y
1
:
m
y
y
x
x
=
−
−
1
2
1
2
6. Solve for
x
1
:
m
y
y
x
x
=
−
−
1
2
1
2
7. Solve for
h
:
A = ½bh
8. Solve for
y
:
x
2
+ y
2
= r
2
Luttrell 2012
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Name: ______________________
Date: _____
5b – More Practice Translating
Algebra was founded by people who needed to find answers to problems. For centuries, people
would try to solve problems without the use of variables. It wasn’t until Fibonacci in the 1400s
made using symbols (mathematical operators and variables) popular. Note the difference
between the two equations below:
Tom’s age plus 4 equals Sally’s age
T + 4 = S
Students typically have problems translating sentences into expressions or equations. Once they
get the equation written, they no longer have as much difficulty.
Translate the following into expressions or equations. Define your variables:
1. Sally is five years less than twice Tom’s age.
2. One less than Sally’s hourly wage.
3. Three times as many rocks
4. Thrice as many hours
5. Sally’s and Tom’s wages sum to be $50,000.
6. Four less pounds
7. The sum of two numbers
8. A number and 3
9. My height is 4 inches more than yours.
10. Five less than a number
11. The height of the room is half the length.
12. Twice the age of Sally.
13. Tom earns 1.25 times as much as Sally.
14. Four more pounds
Solve:
15. I ate ½ my daily peanut butter sandwiches for lunch. Had I eaten one more sandwich, I
would have eaten ⅝ my daily sandwiches. How many sandwiches do I eat daily?
16. Tom wanted Sally’s telephone number. Knowing she lived in Berrien Springs, with an
exchange number of 473, Tom just needed the last four digits. Sally slyly stated that 45 added to
his age of 30 equaled to 1000 less than half her telephone number (ignoring the exchange).
Luttrell 2012
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Name: ______________________
Date: _____
5c – Number Line
Real numbers can be plotted on a line from left to right with the numbers in ascending order.
That means negatives are placed on the left end and positives on the right. Remember -100 is
smaller than -1, so -100 would be further left than -1. For equalities, a solid dark dot on the
number line indicates the value of the variable that makes the equation true. For example, an
equation is found that
x
= 4. The graph of
x
= 4 is shown below:
| | | | | | | |
-2 -1 0 1 2 3 4 5 6
Below are examples of inequalities and their solutions graphed. Remember that the inequality
symbols switch when multiplying (dividing) by a negative. Also an open circle or ) shows a
strict inequality, as in the constant is not included in the solution set. On the other hand, a closed
circle or ] shows that the constant is included.
E
x
ample: 4
x
- 1 < 7
-
x
-
2 ≥ 2
4
x
< 8
-
x
≥ 4
x
< 2
x
≤- 4
| | | |
◯
| | | |
| | | | | | |
-2 -1 0 1 2 3 4 5 6 -6 -5 - 4 -3 -2 -1 0 1
Solve and graph the solution on a number line:
1. 3
x
- 2 = 13
2. 2
x
+ 2 = 12
3. (4 - 2)
x
= 12
4. -
4 ≤
x
-
1 ≤1
5. -3
w
< 12
6. 6(1 -
x
) - 3
x
≤ 12
7. -2
z
> 12
8. 2
y
≥-11
9. 3
x
+ 4 > 5
Luttrell 2012
77
Name: ______________________
Date: _____
5d – Absolute Values
On a real number line, what is the distance between: 5 and 12, -3 and 5, or 4 and 7? We find the
distance by subtracting the smaller value from the larger: 12-5, 5 - (-3), 7 - 4. So what is the
distance between 0 and
x
? It would be written as
x
- 0. But what if we knew
x
was 5 units from
zero? The number could be five less or five more than zero. Then symbolically it would be
written as |
x
- 0| = 5, then
x
is either -5 or 5. The absolute value,
magnitude
, of
x
- 0 (or
x
) gives
the distance, without specifying direction.
The equation |
x
- 1| = 5, can be thought as the distance
x
from 1 is 5 units. So starting on the real
number line at 1, you would count to the right or left 5 units, getting two answers: 6 and - 4.
E
x
pressions with absolute values can get more complicated, so you may want to remember a
certain rule: |
x
y| = |
x
|
⋅
|y|. For e
x
ample: |-
x
| = 3 can be written as |-1|
⋅
|
x
| = 3. Then |-1| really is 1
since the absolute value is asking for the magnitude of -1. So the equation really is 1
⋅
|
x
| = 3,
which gives the answer of -3 and 3.
Simplify each expression and graph the solution.
1. The distance between 4 and 6 is 2.
2. The distance between 5 and 11 is 6.
3. The distance between 7 and -3 is 10.
4. The distance between x and 3 is 5.
5. The distance between x and -2 is 4.
6. The distance between x and 3 is more than 6.
7. The distance between x and -2 is less than 4.
Solve each equation by translating its symbolic meaning first. Graph the solution.
8. | x - 4 | = 2
9. | x - 8| = 3
10. | x + 3| = 5
11. |x + 2| = 6
Luttrell 2012
78
Name: ______________________
Date: _____
5e – Graphing Absolute Values
Sometimes it is quite impossible to simplify absolute equations unless you get rid of them
altogether. That is done only by finding two equivalent equations and solving each. The reason
for two equations is because the expression inside the absolute value could very well be a
positive or negative value. Now if the expression were positive, the absolute values are
redundant and can simply be dropped. But if the expression were negative, the only way to
make it positive like an absolute value would be to negate the expression.
E
x
ample: |-
x
+ 2 | ≤ 5
|
x
- 3 | > 2
+(-
x
+ 2)
≤5 and -(-
x
+ 2) ≤5
x
- 3 >2 or -(
x
- 3) >2
-
x
≤3 and
x
-
2 ≤ 5
x
> 5 or
x
- 3 < -2
x
≥-3 and
x
≤ 7
x
>5 or
x
< 1
| ● | | | | | | | | | ● |
| | |
◯
| | |
◯
|
- 4 -3 -2 -1 0 1 2 3 4 5 6 7 8 -2 -1 0 1 2 3 4 5 6
E
x
plain why
and
is used in the first e
x
ample and
or
in the second. ________________________
_____________________________________________________________________________
Simplify each expression and graph the solution.
1. x = |-2|
2. x = | -5|
3. |x| = 4
4. |x| = 2
5. x < |- 3|
6. x ≥ |-1|
7. |x| ≤3
8. |x| ≥7
Solve each equation by converting into two equivalent equations first. Graph the solution.
9. | x – 3 | = 5
10. | x – 1 | = 4
11. | x – 5 | = 6
12. |x – 2 | = 3
Luttrell 2012
79
Name: ______________________
Date: _____
5f – Solving Absolute Values
Show the two equivalent equations. Solve and graph the solution set.
1. | x - 2| < 3
2. |x-4|> 2
3. |x+7|< 1
4. |x+2| > 3
5. |
x
-
1 | ≥ 5
6. |
x
+ 2 | ≤ 7
7. |
x
-
2 | ≤ 4
8. | 3
x
+ 1 | < 5
9. | 4
x
- 1 | > 3
10. |2 -
y
| < 2
11. | 2x -
4 | ≤ 5
12. | 3x -
6 | ≥1
13. | 2x -
5 | ≤4
14. | x -
12 | ≤ 3
15
. | x + 3 |≤ 2
Luttrell 2012
80
Name: ______________________
Date: _____
5g – Cartesian Coordinate Plane
Rene Descartes came up with a way to systematize giving directions. He took two real number
lines and had them intersecting at zero to form perpendicular angles. At each integer, you can
draw a vertical or horizontal line. After a while you will have formed a grid, with each line
intersecting at integer coordinates (lattice points). The horizontal real number line is commonly
referred to as the
x
-axis and the vertical number line is called the
y
-axis. The point (
x
,
y
) can be
found by moving left/right along the
x
-axis and then from that new point, moving up/down y-
spaces. For e
x
ample the ordered pair (2,-1) would be found by starting at the origin (0,0) and
moving right 2 spaces and down 1 space.
There are other coordinates, such as polar coordinates, but those are for a later course.
1. Draw a coordinate system, label the integers from -10 to 10 on both a
x
es.
2. Draw a dot and label the points R(-2, 4), E(1, 5), S(4, -3) and T(-7,-2).
3. Draw a coordinate system, label the integers from -10 to 10 on both a
x
es.
4. Label the points A(-2,4), B(-1, 2), and C(3, -6). Connect the dots. What does figure does it
look like? How could you tell that it really is that shape?
