Download Solving Equations

Name: _________________________
Algebra 3
Unit 1
Section 1.2:Properties of Real Numbers
Goals
- Students will be able to graph and order real numbers
-Students will be able to identify properties of real numbers
Vocab
Opposite -(Additive Inverse): the opposite or additive inverse of any number a is -a. The
sum of opposites is zero, the additive identity.
Reciprocal -(Multiplicative Inverse): the reciprocal or multiplicative inverse of any nonzero
number a is
. The product of reciprocals is 1, the multiplicative identity.
Real Numbers
Rationals
Irrationals
Real Number
Sets
Integers
Whole Numbers
Natural Numbers
Examples
Name the set the following belong in
#1) 0
#4)
#2) -5
#5).23
#3)
#6)
Pg. 1
Properties of
Real Numbers
Properties of Addition and Multiplication
Property
Addition
Multiplication
Closure
a + b is a real number
ab is a real number
Commutative
a+b=b+a
ab = ba
Associative
(a + b) + c = a + (b + c)
(ab)c= a(bc)
Identity
a + 0 = a,
Inverse
Distributive
0+a=a
a + (-a) = 0
a · 1 = a, 1 · a = a
a · 1/a = 1 , a ≠ 0
a(b + c) = ab + ac
***The distributive property involves both addition and multiplication***
Examples
Ordering
numbers
Examples
What property does the equation illustrate?
#1)
#2) (3 • 4) • 5 = (4 • 3 ) • 5
#3) (2 + 3) + 4 = 2 + (3 + 4)
#4) 3( g + h) + 2g = (3g + 3h) + 2g
#5) 5 + 0 = 5
#6) -4 · 1 = -4
#1)Plot each point on the number line.
#2) Write out the values from left to right (least to greatest)
#1) 1, -4, -3, 0, 2
#2) -1, 1, -9, -8, 2, 3
Pg. 2
#3) 1.5, -2.2, 3.1, 4.45,
,
#4) -10, -2, -1, -6, -4, -3
Examples
#5) 0, 5, 2, -1, 2.1, -4.1
#6)
, -4, 2.7, 0,
Summary
~ If you value falls in the integer set, which other sets does it fall in?
~ How can you tell the difference between commutative and associative
property?
~ Which order do you go with numbers on a number line if you want to go
from least to greatest?
Section Work: Worksheet: Properties and real numbers
Pg. 3
Due: _________________
Section 1-3: Algebraic Expressions
Goals
-Students will be able to evaluate algebraic expressions
-Students will be able to simplify algebraic expressions
Vocab
Evaluate - in an algebraic expressions, substitute a number for each variable in the
Term
Coefficient
Like Terms
Variable
Constant
Writing an
Algebraic
Expression
Examples
expression
-an expression that is a number, a variable, or the product of a number and
one or more variables.
-the value that is in front of the variable
-terms that have the same variables raised to the same powers
- it is a letter that represents a value
- it is the term without a variable
#1)Use key words in the table below
- Be sure to switch the order of the values and variables when
more than or less than is used
Adding
More than (switch)
Increased
Sum
Plus
Also
And
Subtraction
Less than (switch)
Decreased
Difference
Take away
Multiplication
Product
Times
Double
Division
Quotient
Divided by
Cut in _____
Write a variable expression for each word phrase
#1) 2 more than p
#2) y plus 4
#3) the quotient of 3 divided by y
#4) u divided by 9
Pg. 4
Examples
Write a variable expression for each word phrase
#5) v less than 2
#6) The difference of 4 and k
#7) 2 times the sum of 7 and b
#8) You had $150, but you are spending $2 each day.
#9) You can talk 500 minutes a month and you talk two hours each week.
#1) Plug in the values for each variable into parenthesis
Evaluating
Expressions #2) Evaluate/solve using order of operations
Examples
Evaluate each expression for the given values of x and y.
#1) x4 + 3y when x = 2 and y = -8
#2) -3x2 - 5x + 7 when x = -2
#3) 3x -
#4) x2 + 7y - 2xy when x = -2 and y = 2
when x = -3 and y = 2
Evaluate the expression when a = -3, b = 2, c = -6, and d = -20
#5) a - b - c - d
#6) d2 - 3c
#7) 2ac - d ÷ b
Pg. 5
Combining
Like Terms
Examples
#1) Underline the like terms
#2)Add or subtract the coefficients
- Do not change the exponents unless you are multiplying
Combine all like terms
#1) 3x + 2y +12x
#2) 9x - 3xy + 7xy - 7x
#3) 13x2 - 3x2 + 5x - 2x
#4) 4rs - 2r - 2s + 8rs
#5) 5x(x + 6)
#6) 4x( 3x - 8)
#7) -5x(1 - 3x)
#8) -4x(x2 - 3x)
#9) 6(x - y) - 4(x - y)
#10) 7x - (9x + 5)
Summary
~ What phrase tells you to switch the order of the values when you write it
out?
~ When do you change the exponents when you are combining like terms?
Pg. 6
Section Work: Worksheet: Algebraic Expressions
Due: _________________
Section 1-4: Solving Equations
Goals
- Students will be able to solve equations
- Students will be able to solve literal equations
Vocab
Equation - a statement that two expressions are equal
Inverse Operations - operations that undo each other
Literal Equations - an equation that uses at least two variables
Solving
Equations
Examples
#1) Box the variable
#2) The whole goal is to get the variable on its own side
- multiply into parenthesis to get rid of them , combine like terms
- add or subtract to move anything away
- multiply or divide to get the variable alone
#1) 5 + m = -10
#2) 12 - x = 7
#3)
#4) 14n = 112
#5) 4a + 16 = 48
#6)
Pg. 7
Examples
Story
Problems
Examples
#7) 4x - 2 = 3x + 4
#8) 5x - 6 = 6x + 16
#9) 5x - (4x + 2) = 3
#10)4m + 1 + m = 3m + 9
#11) 3p +2(4p - 6) = 10
#12) 2x -2(2x + 1) = -5
**** Remember the key words to look for.
#1) The sides of a triangle are in the ratio 5: 12: 13. What is the length of
each side if the perimeter is 150?
Pg. 8
Examples
#2) Two trains left a station at the same time. One traveled north at a certain
speed and the other traveled south at twice that speed. After 4 hours, the trains
were 600 miles apart. How fast was each train traveling?
#3) You and your friend left a bus terminal at the same time and traveled in
opposite directions. Your bus was in heavy traffic and had to travel 20 miles
per hour slower than your friend’s bus which was going 60 miles per hour.
When will they be 200 miles apart?
Literal
Equations
Examples
#1) Box the variable to solve
- move everything away from that variable like before
Solve for the indicated variable.
#1) x - y = z: for y
#2) 2p = 3q - 4: for q
#3) 7x - 3y = 8 for y
#4) x - 5 = y for x
Pg. 9
Examples
Solve for the indicated variable.
#3) S = L(1 - r); for r
#4) A= lw + wh + km; for w
#5) a = xyz; for y
#6) yz= 2(3x - 4y); for y
#7) a =
bcd; for c
Summary
~ What is the first step in solving an equation after you box the variable?
~ What is the second thing you do when solving an equation?
~What is the last step in solving an equation?
Section Work: Worksheet: Equations
Pg. 10
Due: _________________
Section 1-5: Solving Inequalities
Goals
- Students will be able to solve and graph inequalities
- Student will be able to solve compound inequalities
Vocab
Compound - joining two inequalities with the word and or the word or.
inequality
#1) Is the variable negative ===> yes - then you need to switch the in
Solving
equality sign at the end of the problem
and graphing
inequalities #2) Solve for the variable
- answer must be in the form “variable...symbol...value”
#3)Graph
- O (open circle) for < or > signs
- ( closed circle) for ≤ or ≥ signs
Examples
Examples:
Solve each inequality and solve.
#1) -n + 4 > 8
#2) -5 > 2x + 13
#3) -5x + 9 ≤ 2(x - 6)
#4) -3(2x - 5) -1 ≥ 4
Pg. 11
Solving an
“AND”
compound
inequality
Examples
#1) Separate if the inequalities are pushed into one
#2) Solve each one separately
#3) Graph each statement
- where they overlap is the answer (And = BOTH)
#1) 7 < 2x + 1 and 3x ≤ 18
#2) 5 ≤ 3x - 1 and 2x < 12
#3) -3 ≤ -2x ≤ 6
#4) -4 ≤ -x + 1 < 6
#5) 3x ≤ -12 and 5x ≥ 5
#6) -33 ≤ -7n - 12 < -26
Pg. 12
Solving an “OR”
compound
inequality
#1) Solve each inequality separately
#2) Graph each solution
- they will go apart or cover the whole number line
#1) 2x - 5 ≥ 1 or 2x - 5 ≤ -1
#2) 5x + 6 ≤ 11 or -3x ≤ -12
#3) x > 0 or 5x - 4 < -14
#4) p + 4 ≤ 1 or p - 1 ≥ 1
#5) 12 + 4n > 44 or 10 - 12n> -38
#6) 18 ≥ 7x + 4 or 5x - 15 < 5
Summary
~ What is the direction (s) that “AND” inequalities can go?
~ What is the direction (s) that “OR” inequalities can go?
Pg. 13
Section Work: Worksheet ~ Compound inequalities
Due: _________________
Section 1-6: Absolute Value Equations and Inequalities
Goals
- Student will be able to solve equations and inequalities involving absolute value
Vocab
Absolute Value - the distance from zero on the number line |x|
Extraneous
Solutions - Solutions obtained from solving a problem that are not true answers. They
are considered false solutions.
#1) Solve for the | |
Solving
#2) Rewrite as two equations
an absolute
-one is = to a positive answer
value equation
- one is = to a negative answer
#3) Solve each equation separately
#4) Check your answers for extraneous solutions
**** The absolute value equation | ax + b | = c, where c > 0, is equivalent to the
compound statement ax + b = c OR ax + b = -c ****
Examples
Pg. 14
#1) | x + 5 | = 7
#2) 3|x + 2 | - 1 = 8
#3) | 3x + 2 | = 4x + 5
#4) -8 | -2n | + 5 = -75
#5) 2| -2r - 1 | = 22
#5)| 5x - 2 | = 7x + 14
Solving
an inequality of
the form
|ax + b | < c
(<) ---- AND
#1) Solve for the | | then you need to separate into two problems
- one positive < +
- one negative > #2)Solve each separately
#3)Graph each answer on one number line
**** The inequality | ax + b | < c, where c > 0, means that ax + b is between -c
and c. This is equivalent to -c < ax + b < c ****
* < can be replace by ≤ ( and statement)*
Examples
#1) | 2x + 3 | < 7
#2) | x - 5 |+ 1 ≤ 3
#3) | p + 4 | ≤ 8
#4) |x - 2 | - 5 < -2
Pg. 15
Solving
an inequality of
the form
|ax + b | > c
(>) ---- OR
#1) Solve for the | | then you need to separate into two problems
- one positive > +
- one negative < #2) Solve each separately
#3) Graph on one number line
**** The inequality | ax + b | > c, where c > 0, means that ax + b is beyond -c
and c. This is equivalent to ax + b < -c or ax + b > c****
* > can be replace by ³ ( and statement)*
#1) | 1 - 4k | ≥ 11
#2) | 2y + 4 | - 2 > 6
#3) | t + 2 | ≥ 10
#4) | 4v - 7 | + 8 > 17
Examples
Summary
~ Is an absolute value inequality that goes < an AND or OR?
~ Is an absolute value inequality that goes > an AND or OR?
~ How many answers do you have with absolute value equations?
Pg. 16
Due: _________________
Section Work: Worksheet ~ Absolute value equations and inequalities