Transcript Chapter 3

Intermediate Algebra Chapter 3
•Linear Equations
•and
•Inequalities
Denis Waitley
• “Failure should be our teacher, not
our undertaker. Failure is delay,
not defeat. It is a temporary
detour, not a dead end. Failure is
something we can avoid only by
saying nothing, doing nothing, and
being nothing.”
Intermediate Algebra 3.1
•Introduction
•To
•Linear Equations
Def: Equation
• An equation is a
statement that two
algebraic expressions
have the same value.
Def: Solution
• Solution: A replacement for the
variable that makes the equation
true.
• Root of the equation
• Satisfies the Equation
• Zero of the equation
Def: Solution Set
• A set containing all the
solutions for the given
equation.
• Could have one, two, or many elements.
• Could be the empty set
• Could be all Real numbers
Def: Linear Equation in One
Variable
• An equation that can be written in
the form ax + b = c where a,b,c are
real numbers and a is not equal to
zero
Linear function
• A function of form
• f(x) = ax + b where a and b
are real numbers and a is not
equal to zero.
Equation Solving: The Graphing
Method
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1. Graph the left side of the equation.
2. Graph the right side of the equation.
3. Trace to the point of intersection
Can use the calculator for intersect
The x coordinate of that point is the solution
of the equation.
Equation solving - graphing
• The y coordinate is the value of both the left
side and the right side of the original
equation when x is replaced with the
solution.
• Hint: An integer setting is useful
• Hint: x setting of [-9.4,9.4] also useful
Def: Identity
• An equation is an identity if every
permissible replacement for the variable is a
solution.
• The graphs of left and right sides coincide.
• The solution set is R
R
Def: Inconsistent equation
• An equation with no solution is an
inconsistent equation.
• Also called a contradiction.
• The graphs of left and right sides never
intersect.
• The solution set is the empty set.

Example
1
x  19  2 x  6
2
Example
x  3  1 x
Example
x 3  3 x
Def: Equivalent Equations
• Equivalent equations are equations that
have exactly the same solutions sets.
• Examples:
• 5 – 3x = 17
• -3x= 12
• x = -4
Addition Property of Equality
• If a = b, then a + c = b + c
• For all real numbers a,b, and c.
• Equals plus equals are equal.
Multiplication Property of
Equality
• If a = b, then ac = bc is true
• For all real numbers a,b, and c
where c is not equal to 0.
• Equals times equals are equal.
Solving Linear Equations
• Simplify both sides of the equation as
needed.
– Distribute to Clear parentheses
– Clear fractions by multiplying by the LCD
– Clear decimals by multiplying by a power of 10
determined by the decimal number with the
most places
– Combine like terms
Solving Linear Equations Cont:
• Use the addition property so that all variable
terms are on one side of the equation and all
constants are on the other side.
• Combine like terms.
• Use the multiplication property to isolate
the variable
• Verify the solution
Ralph Waldo Emerson – American essayist,
poet, and philosopher (1803-1882)
• “The world looks like a
multiplication table or a
mathematical equation,
which, turn it how you
will, balances itself.”
Useful Calculator Programs
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CIRCLE
CIRCUM
CONE
CYLINDER
PRISM
PYRAMID
TRAPEZOI
APPS-AreaForm
Robert Schuller – religious leader
• “Spectacular achievement
is always preceded by
spectacular preparation.”
Problem Solving 3.4-3.5
• 1. Understand the Problem
• 2. Devise a Plan
– Use Definition statements
• 3. Carry out a Plan
• 4. Look Back
– Check units
Les Brown
• “If you view all the things
that happen to you, both
good and bad, as
opportunities, then you
operate out of a higher level
of consciousness.”
• Albert Einstein
»“In
the middle of
difficulty lies
opportunity.”
Linear Inequalities – 3.2
• Def: A linear inequality in one
variable is an inequality that can be
written in the form ax + b < 0
where a and b are real numbers and
a is not equal to 0.
Solve by Graphing
• Graph the left and right sides and find the
point of intersection
• Determine where x values are above and
below.
• Solution is x values – y is not critical
Example solve by graphing
15  x  x  1
15  x  x  1
Addition Property of Inequality
• If a < b, then a + c = b + c
• for all real numbers a, b, and c
Multiplication Property of
Inequality
• For all real numbers a,b, and c
• If a < b and c > 0, then ac < bc
• If a < b and c < 0, then ac > bc
Compound Inequalities 3.7
• Def: Compound
Inequality: Two
inequalities joined by
“and” or “or”
Intersection - Disjunction
• Intersection: For two sets A and B, the
intersection of A and B, is a set containing
only elements that are in both A and B.
A B
Solving inequalities involving
and
• 1.
Solve each inequality in the
compound inequality
• 2. The solution set will be the
intersection of the individual
solution sets.
Union - conjunction
• For two sets A and B, the union of
A and B is a set containing every
element in A or in B.
A B
Solving inequalities involving
“or”
• Solve each inequality in the
compound inequality
• The solution set will be the union
of the individual solution sets.
Confucius
• “It is better to light one
small candle than to
curse the darkness.”
Absolute Value Equations
• If |x|= a and a > 0, then
• x = a or x = -a
• If |x| = a and a < 0, the
solution set is the empty set.
Procedure for Absolute Value
equation |ax+b|=c
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1. Isolate the absolute the absolute value.
2. Set up two equations
ax + b = c
ax + b = -c
3. Solve both equations
4. Check solutions
Procedure Absolute Value
equations: |ax + b| = |cx + d|
• 1. Separate into two equations
• ax + b = cx + d
• ax + b = -(cx + d)
• 2. Solve both equations
• 3. Check solutions
Inequalities involving absolute
value |x| < a
• 1. Isolate the absolute value
• 2. Rewrite as two inequalities
• x < a and –x < a (or x > -a)
• 3. Solve both inequalities
• 4. Intersect the two solutions note the use
of the word “and” and so note in problem.
Inequalities |x| > a
• 1. Isolate the absolute value
• 2. Rewrite as two inequalities
• x>a
or
–x > a
(or x < -a)
• 3. Solve the two inequalities – union the
two sets **** Note the use of the word “or”
when writing problem.
Joe Namath - quarterback
• “What I do is prepare
myself until I know I
can do what I have to
do.”
Intermediate Algebra 3.6
•Graphs
•Of
•Linear Inequalities
Def: Linear Inequality in 2
variables
• is an inequality that can be
written in the form
• ax + by < c where a,b,c are
real numbers.
• Use < or < or > or >
Def: Solution & solution set
of linear inequality
• Solution of a linear inequality
in two variables is a pair of
numbers (x,y) that makes the
inequality true.
• Solution set is the set of all
solutions of the inequality.
Procedure: graphing linear
inequality
• 1. Set = and graph
• 2. Use dotted line if strict inequality or
solid line if weak inequality
• 3. Pick point and test for truth –if a
solution
• 4. Shade the appropriate region.
Linear inequalities on calculator
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Set =
Solve for Y
Input in Y=
Scroll left and scroll through icons
and press [ENTER]
• Press [GRAPH]
Calculator Problem
4
y
x2
5
Compound Inequalities
• Graph both inequalities
• AND – Intersection of
both sets
• OR – Union of both sets.
Abraham Lincoln U.S. President
•“Nothing valuable
can be lost by
taking time.”