Download Advanced Algebra Chapter 1

Advanced Algebra Chapter 1
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(1-1) Evaluating Expressions
What does PEMDAS stand for?
Evaluate the expressions below!
(ex 1) 20 ÷ [1 + (3)2]
(ex 2) 17 – (6 – 9)3 + 23
(ex 3) 24 - 32
5-12
If a = –2, b = 2.5, and c = 3, solve the
following expressions.
(ex 4) a3 + b(c – 1)2 – c2
(ex 5) ab – [c + (2b – a2)]
(ex 6) The formula for the orbital
period T of a satellite is
T = 2πr
v
where, r is the radius of the orbit of
the satellite, and v is the velocity of
the satellite.
Find the period of a satellite in orbit
above Earth if the radius of the orbit
is 4268 miles and the velocity is 4.4
miles per second. Express the orbital
period in hours.
4,268 miles = r
4.4 miles/second = v
T = 2πr = 2(3.14)(4,268) = 26,803.04
v
4.4
4.4
T = 6,091.6 seconds/60 seconds
T = 101.53 minutes/60 minutes
T=
1 min = 60 sec, 1 hour = 60 min. 1.692
hours
1 min = 60 sec, 1 hour = 60 min.
(1-2) Simplifying Expressions
What do the directions “simplify”
mean?
- Make the expression smaller
- Combine like terms
Simplify each expression below.
(ex 1) 5x – 3y – 2x + 3y
5x – 2x -3y +3y
3x + 0 = 3x
(ex 2) 4c – 2c – (5c + c)
4c – 2c – 5c – c =
2c – 5c – c =
-3c – c = -4c
(ex 3) 3(r – 10) – 4(7s + 2r)
(ex 4) ½ (4w – 8y) + ¼ (12w + 4y)
2w – 4y + 3w + y
2w + 3w = 5w
-4y + y = -3y
5w – 3y
Solving Equations (1-3)
Review: What is opposite of the
following operations?
Addition 
Subtraction 
Multiplication 
Division 
Def: isolate
Steps to Solve:
1) Circle the variable
2) Determine operations
connected to variable
3) Cancel using opposite
operation
4) Solve to isolate variable
5) Check!
Solve each equation!
(ex 1) 5x = 45
(ex 2) 17 = a – 6
(ex 3) 0.2m = -10
(ex 4) ¼ m = -6
Def: Multi-Step Equation
When solving equations, what
method can we use to determine
what to cancel first?
An equation that requires more
than one step.
SADMEP
Steps to Solve:
1) Circle the variable
2) Determine operations
connected to variable
3) Cancel using opposite
operation using SADMEP
4) Solve to isolate variable
5) Check!
Solve each equation!
(ex 5) 3x + 4 = 10
3x + 4 = 10
-4 -4
3x = 6
3
3
x=2
Solve the equations below!
(ex 6) ½y – 3 = -5
x by ½
- by 3
½y – 3 = -5
+3 +3
(2) ½y= -2(2)
y= -4
(ex 7) 5(4 – 2k) = -5
20 – 10k = -5
-20
-20
-10k = -25
-10 -10
k= 2.5
(ex 8) 5t – 9 = 4t + 1
5t – 9 = 4t + 1
-4t
-4t
t–9=1
+9 +9
t = 10
Solve each equation below!
(ex 9) 3x + 17 = 5x – 13
3x + 17 = 5x – 13
- 17
-17
3x = 5x - 30
-5x -5x
-2x = -30
-2
-2
x = 15
(ex 10) 2.2n + 0.8n + 5 = 4n
3n + 5 = 4n
-3n
-3n
5=n
n=5
Jason started with d dollars in his
d=?
piggy bank. One week later, Jason
doubled the amount in his bank.
2(d) + 20 = 50
Another week later, Jason was able
-20 -20
to add $20 to his bank. At this point,
2d = 30
the piggy bank has $50, what is d?
2
2
d = $15
Absolute Value Equations (1-4)
Def: Absolute value
The number of places a point is from
zero
Real-World Examples:
Distance, time, money (in dialogue)
*Remember to take the absolute
value when the only thing in the bars
is a single number 
Evaluate each expression if w = -4,
x = 2, y = ½ and z = -6
(ex 1) | 3x – 9|
(ex 2) 5 + |z + w|
(ex 3) 14 – 2|w - xy|
| 3x – 9|
|3(2) – 9|
|6 – 9| = |-3| = 3
5 + |z + w|
5 + | (-6 + - 4)|
5 + |-10| = 5 + 10 = 15
14 – 2|w - xy|
14 – 2|-4 – (2)(.5)|
14 – 2|-4 -1|
14 – 2|-5|
14 – 2(5)
14 – 10 = 4
Evaluate each expression if w = -4,
x = 2, y = ½ and z = -6
(ex 4) |wz| - |wy|
(ex 5) |2 – 2w| - 3|xz|
Barack Obama’s approval rating as
President is stated as 54% with a
margin of error of ±4%. Determine
the range of Obama’s approval
rating.
|wz| - |wy|
|(-4)(-6)| - |(-4)(0.5)|
|24| - |-2|
24 – 2 = 22
|2 – 2w| - 3|xz|
|2 – 2(-4)| - 3|(2)(-6)|
|2 + 8| - 3|-12|
|10| - 3|-12|
10 – 3(12)
10 – 36 = -26
54% + 4% = 58 %
54% - 4% = 50%
50% - 58%
Solving Absolute Value Equations
Steps to Solve:
1) Write problem without bars
2) Solve for variable
3) Write problem without bars
and make right side negative
4) Solve for variable
5) Check!
| a – 4 | = 12
Solve each equation!
(ex 1) | a – 4 | = 12
(ex 2) | 2y -3 | = 29
a – 4 = 12
+4 +4
A= 16
2y – 3 = 29
+3 +3
2y = 32
2 2
y = 16
y = {-13, 16}
a – 4 = -12
+4 +4
a = -8
2y – 3 = -29
+3 +3
2y = -26
2
2
y = -13
Solve each equation!
(ex 3) 7| x + 3| = 42
7| x + 3| = 42
7
7
|x + 3| = 6
x+3=6
-3 -3
x=3
x + 3 = -6
-3
-3
x = -9
x = {-9, 3}
(ex 4) -2 |4x – 1| = -24
-2 |4x – 1| = -24
-2
-2
| 4x – 1 | = 12
4x – 1 = 12
+1 +1
4x = 13
4
4
x = 3.25
x = {-2.75, 3.25}
4x – 1 = -12
+1 +1
4x = - 11
4
4
x = -2.75
Solving Inequalities (1-5)
Def: inequality
Review:
Greater than – >
Less than –
<
Greater than or equal to –
Less than or equal to –
What’s the difference between these
symbols?
in = NOT
equality = equal
- symbols that represent
relationships of numbers and
variables that are not equal
2 > 2 False
2 >= 2 True
Special Rule for Inequalities
(ex) 7 > 5
7 > 5 True
7(-1) > 5(-1)
-7 > -5 FALSE
- 7 < - 5 – we must flip the sign 
*Whenever we multiply or divide by
negative numbers, flip the inequality
Solve the inequality then graph!
(ex 1) x + 2 > 5
x+2>5
- 2 -2
x>3
Solve the inequality, then graph!
(ex 2) -5y < -25
-5y < -25
-5 -5
y > 5
(ex 3) m ≥ 7
-2
(ex 4) 4a + 12 ≤ -8
m≥7
-2
(ex 4) 4(5x + 7) < 13
Define a variable and write an
inequality for each problem then
solve.
(ex 5) nineteen more than a number
is less than 42
(ex 6) One half of a number is more
than 6 plus the same number.
Jan’s account balance is $3800. Of
this, $750 is for rent, $59 for her cell
phone, and $220 for food. Jan wants
to keep a balance of $500 in her
account. Write an inequality
describing how much she can
withdraw and still have enough
money.
Absolute Value Inequalities (1-6)