Download Section 2.2 – Simplifying Algebraic Expressions

Section 2.2 – Simplifying Algebraic Expressions
Definitions
a) Commutative Law for Addition
b) Commutative Law for Multiplication
c) Associative Law for Addition
d) Associative Law for Multiplication
e) Distributive Law
f) Equivalent Expressions
g) Definitions: Term
h) Definitions: coefficient
i) Definition: Like Terms
j) Definition: Unlike terms
k) Definition: Combine like terms
l) Definition: Simplify Expressions
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Example #1
Simplify the following expressions. Please justify your steps
a) 6( x + 7 )
b) − 4(− 9 x )
c) − 5(2 x − 3) + 7
d) 6.4 + 3.9(6.1x − 4.4 )
e) − 4 y (3 y − 4 )
f) r 2r 2 − t
(
)
Example #2
a) Simplify 7 − 4( x + 2 )
b) Evaluate 7 − 4( x + 2 ) at x = 3
c) Evaluate the expression you found in part a) at x = 3
d) What does this tell you?
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Example #3
Simplify the following expression. Justify your steps
2(7 x − 5 y ) − 5(3 x + y )
Example #4
Simplify the following expression. Justify your step
a) 16t 3 + 4t − 5t + 3t 3
b) 8 x − (3 x − 7 )
c) − 6 x( x − 1) + x 2
d)
⎛a⎞ ⎛a⎞ 3 1
e) 16⎜ ⎟ − 6⎜ ⎟ + −
⎝b⎠ ⎝b⎠ 2 2
f)
2
(12 x + 15) + 16
3
3t [4 − (t − 3)] + t (t − 5)
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Section 2.3 – Algebra and Problem Solving
Example #1: Translate the English phrase to a mathematical expression.
a) The number minus the product of the number and 3
b) -6 times the sum of the number and 4
c) Three times the difference of a number and 5
d) The square of a number, increased by 1
e) The square of x decreased by the product of x and 2x
Example #2
Write a verbal description of the algebraic expression, without using variables.
a) x − 10
4
b) 5 x + 4
c)
1 t
−
2 5
Example #3
a) A cash register contains d dimes. Write an algebraic expression that represents the
total amount of money (in dollars)
b) A campground charges $15 dollars for adults and $2 for children. Write an algebraic
expression that represents the total camping fee for the m adults and n children.
c) Each share of a certain stock is worth $5. Complete the table below, and find an expression
that describes the total value (in dollars) of n shares of the stock.
Number of
Shares
1
2
3
4
Total Value (dollars)
n
5
Section 2.4 – Introduction to Equation
Definitions
a) Equation
b) Solution to an equation
c) Equivalent Equations
d) Addition Property of Equality
e) Subtraction Property of Equality
f) Multiplication Property of Equality
g) Division Property of Equality
h) Interchange the two sides of Equation
Example #1
Determine whether each value of x is a solution of the equation.
6 x + 1 = −11
a) x = 2
b) x = −2
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Example #2
Determine whether each value of x is a solution of the equation.
3(3 x + 2 ) = 9 − x
a) x = −
3
4
b) x =
3
10
Example #3
Determine whether each value of x is a solution of the equation.
x2 − 4 = x + 2
a) x = 3
b) x = −2
Example #4
Determine whether each value of x is a solution of the equation.
4 2
+ =1
x x
a) x = 0
b) x = 6
7
Example #5
Justify each step of the solution
6 x = 17
6 x 17
=
6
6
17
x=
6
Example #6
Justify each step of the solution
2( x − 1) = x + 3
2x − 2 = x + 3
2x − 2 − x = x + 3 − x
x−2=3
x −2+ 2 = 3+ 2
x=5
Example #6
Justify each step of the solution
x
= x +1
3
⎛x⎞
3⎜ ⎟ = 3( x + 1)
⎝3⎠
x = 3x + 3
x − 3x = 3x + 3 − 3x
x − 3x = 3x − 3x + 3
− 2x = 3
− 2x
3
=
−2 −2
3
x=−
2
Example #7
Write a verbal description of the algebraic equation without using variables.
a) 3 x − 2 = 7
b) 10( x − 3) = 8 x
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Example #8
Write an algebraic equation. DO NOT SOLVE THE EQUATION
a) The difference of 24 and a number is 3
b) Four times the sum of a number and 6 is 100
c) Four divided by the sum of a number and 5 is 2
Example #9
Simplify the expression. State the units of the simplified value.
⎛ 3 dollars ⎞
⎟(5 units ) =
⎝ unit ⎠
a) ⎜
⎛ 5 ft ⎞⎛ 60 sec ⎞
⎟⎜
⎟(20 min ) =
⎝ sec ⎠⎝ min ⎠
b) ⎜
⎛ $12 ⎞⎛ 1hr ⎞
⎟⎜
⎟(45 min ) =
⎝ hr ⎠⎝ 60 min ⎠
c) ⎜
Example #10 – Write an algebraic equation. DO NOT SOLVE IT!!!
A student has n quarters and seven $1 bills totaling $8.75. How many quarters does the student
have?
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