Chapter 2
Linear Equation and Inequalities
Michael Giessing
giessing@math.utah.edu
University of Utah
Linear Equation and Inequalities – p.1/12
Forming Equivalent Equations
An equation can be rewriten into an equivalent form
by:
1. Simplify either side
2. Using the Golden Rule of Algebra
3. Interchanging sides
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The Golden Rule (of Algebra)
Do unto one side as thou hath done unto the
other.
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Examples
Simplify the left hand side.
x(4 − x) = 2x →
4x − x2 = 2x
Added 4 to both sides.
x(4 − x) = 2x →
x(4 − x) + 4 = 2x + 4
Interchangesd sides.
x(4 − x) = 2x →
2x = x(4 − x)
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The Golden Rule and Addition
You can think of an equation as a scale. If you do
something to one side you must do the same thing to
the other side to keep things in balance. You must do
all operations to entire expressions!
You can add anything to both sides. This includes
negative numbers and variables
y+π
y + π + (−3)
2x + 3
2x + x
=
=
=
=
y2 →
y 2 + (−3)
x→
−x + x
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The Golden Rule and Multiplication
You can mulitiply both side by any number. You must
multiply the entire side by the number not just some
of the terms on that side.
2l + 6 = 4l → 30(2l + 6) = 30 × 4l
You can multiply by any multiplicative inverse.
Which is the same thing as saying you can divide by
any number.
4
1 4
1
= 5n → × = × 5n
n
4 n 4
There is a number that has no multiplicative inverse.
What is it?
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Never divide by Zero
Zero does not have a multiplicative inverse. To see
this try to find Zero’s inverse.(Remember that an
inverse is the number x such that ax = 1. For most
numbers x = 1/a)
0∗x=1?
Unsual things happen if you divide by zero.
Linear Equation and Inequalities – p.7/12
The Golden Rule and Multiplication by a variable
You can alway multiply both sides by a variable.
x2 + y 2 = 4 → x(x2 + y 2 ) = 4x
You can divide by x as long as x 6= 0. It is easy to
make this mistake. Be cautious whenever you divide
by a variable.
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The Golden Rule and Function
We will use the Golden Rule with more than just addition and multiplication. We will use it with many other
operations called functions. Always remember to apply the operation to the entire left hand side(LHS) and
right hand side( RHS).
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Word Problems
1. What do you need to know to solve the problem?
Write this down in english.
2. Assing numbers to the the known parts. Assign a
letter to the unknown parts
3. Translate this into an algebraic equation or
inequality.
4. Solve.
5. Make sure that your solution answers the original
question
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Example
You purchased my new bike for $1500.00. The bike
shop was having 30% off sale the day you purchased
the bike. What was the original price of the bike?
1. The price you paid for the bike is 30% off the
original price. Restated, the price you paid for the
bike is the orignal price minus 30% of the
original price.
2. 1500.00 =x-.3x
1500.00=.7x
Simplifying the RHS
1500.00/.7=.7x/.7 Golden Rule of Algebra
3.
2142.86=x
Simplifying both sides
x=2142.86
Interchanging the sides
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Example Continued
The answer $2142.86 answers the question and seems
reasonable. Lets check it to make sure everything
worked. We need to see if 30% off of $2142.86 is
$1500.00.
2142.86 − .3 × 2142.86 = 1500.00
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Useful Formulas
Business
Selling price = Cost + Mark up
Mark up = Mark up rate × Cost
Selling price = List price - Discou
Rates in Mixtures
first rate × Amount + second rate
= Final Rate × Amount
Rate problems
Distance = rate × time
Linear Equation and Inequalities – p.13/12
Geometry Formulas
Shape
Square
Rectangle
Circle
Triangle
Area
A = s2
A = lw
A = πr 2
A = bh/2
Perimeter
P = 4s
P = 2l + 2w
P = 2πr
P =a+b+c
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3-D Geometry Formulas
Shape
Cube
Rectangular Prism
Cylinder
Sphere
Volume
V = s3
V = lwh
V = πr 2 h
A = 43 πr3
Surface Area
SA = 6s2
SA = 2lw + 2wh +
SA = 2πrh + 2πr 2
SA = 4πr 2
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