Download MTH 60 Elementary Algebra I

MTH 070
Elementary Algebra
Chapter 2
Equations and Inequalities in
One Variable with Applications
2.1 – Properties of Equality and
Linear Equations
Copyright © 2010 by Ron Wallace, all rights reserved.
Three Algebraic Processes



Changing expressions into simpler
equivalent expressions.
Changing equations/inequalities into
simpler equivalent equations/inequalities.
Solving equations/inequalities.
Primary
Primary
(i.e. only)
Tools?
Tools?
The algebraic properties.
“Don’t make stuff up.”
Review of Basic Properties




Identities (0 & 1)
These are
used to find
Inverses
simpler
 Opposites: a+(-a) = 0
equivalent
 Reciprocals: b*(1/b) = b/b = 1
expressions.
Commutative

x + 0 = x & 1x = x

Order of calculations (+ or *)
Associative


Grouping of calculations (+ or *)
Distributive
Solving Equations

Solution or Root : A value for the
variable (or set of values for the variables)
that makes the equation a true statement.


How many solutions?
Finding Solutions:




Guess!
Estimate
Graphically
Algebraically
2x  5  9
x y 5
x 1  5
2
This chapter …
Linear Equations w/ 1 Variable

With grouping symbols removed …

Linear terms


Constant terms


Example: 7x
Example: 12
One variable

Possibly more than once.
3( x  2)  9  4 x
2x  5  9
x y 5
x 1  5
2
Solving Algebraically

Basic Strategy …



Linear
Equations
w/
1 Variable
Determine simpler “equivalent equations.”
Goal: x = ?
Tools …




Arithmetic
Basic Properties
Addition Property of Equality
Multiplication Property of Equality
Expressions VS. Equations

Expressions only involve constants,
variables, operators, & grouping symbols.


Simplify & Evaluate
An Equation is a statement that two
expressions are equal.

Solve
3( x  2)  9  4 x
3( x  2)
9  4x
Symmetric Property of Equality
a

b
Symmetric Property of Equality
b

a
Symmetric Property of Equality
a

b
b

Switching sides of an equation
produces an equivalent equation.
a
Addition Property of Equality
a

b
Addition Property of Equality
ac

b
Addition Property of Equality

Any expression may be added to both
sides of an equation without changing its
solutions.
ac
b

c

Solving:

Add the opposite of
equation.
 Why?
5

x+5=7
+ (-5) = 0
5 to both sides of the
and
x+0=x
Similar forms …
5+x=7
7=x+5
x–5=7
5–x=7
Solving:

x + [something] = [something else]
A different but equivalent viewpoint …

A term may be moved to the other side of an
equation by changing its sign.
x+5=7
x=7–5
x=2
Solving:

x – [something] = [something else]
A different but equivalent viewpoint …

A term may be moved to the other side of an
equation by changing its sign.
x–5=7
x=7+5
x = 12
Multiplication Property of Equality
a

b
Multiplication Property of Equality
ac  b
Multiplication Property of Equality

Both sides of an equation may be
multiplied by any non-zero expression
without changing its solutions.
ac  bc
Why?
Solving:

Multiply both sides of the equation by the
reciprocal of
5.

or … divide both sides by 5.

Why?
5

5x = 15
(1/5) = 1
and
1x = x
Similar forms …
x
 15
5
2
3
x5
2x
5
3
5x  15
Checking a Solution

In the original equation, replace all occurrences
of the variable by the solution.


Note: Put the solution in parentheses.
Independently determine the value of each side
of the equation.



i.e. Do the arithmetic.
Use a calculator (esp. TI-84).
Compare the results … are they the same?
ALWAYS check your solutions!
Using Both Properties
2x  7  15
2x 12  5x
2x  7  x  5
2 x  7  7  5x
Example 1 of 2
Applications

A rhombus is a 4-sided figure with all sides having the
same length. In Europe there is a rhombus-shaped sign
that indicates a priority road. Find the length of each
side of the priority road sign if its perimeter is 243.84
centimeters.
Example 2 of 2
Applications

Dean Silver has been offered two commission-based jobs
selling motorcycles for two local dealers. One dealer is
offering 2% commission on total monthly sales plus
$1,800 per month, while the other dealer is offering 4%
commission on total monthly sales plus $1,200 per
month. Solve the equation 0.02s + 1800 = 0.04s + 1200,
where s represents total monthly sales, to determine how
much Dean must sell so that the wages of the two jobs
are equal.