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Intermediate Algebra Chapter 3 • Linear Equations • and • Inequalities

Intermediate Algebra Chapter 3

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Page 1: Intermediate Algebra Chapter 3

Intermediate Algebra Chapter 3

•Linear Equations

•and

•Inequalities

Page 2: Intermediate Algebra Chapter 3

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.”

Page 3: Intermediate Algebra Chapter 3

Intermediate Algebra 3.1

•Introduction

•To

•Linear Equations

Page 4: Intermediate Algebra Chapter 3

Def: Equation

•An equation is a statement that two algebraic expressions

have the same value.

Page 5: Intermediate Algebra Chapter 3

Def: Solution

• Solution: A replacement for the variable that makes the equation true.

• Root of the equation• Satisfies the Equation• Zero of the equation

Page 6: Intermediate Algebra Chapter 3

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

Page 7: Intermediate Algebra Chapter 3

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

Page 8: Intermediate Algebra Chapter 3

Linear function

• A function of form

• f(x) = ax + b where a and b are real numbers and a is not equal to zero.

Page 9: Intermediate Algebra Chapter 3

Equation Solving: The Graphing Method

• 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.

Page 10: Intermediate Algebra Chapter 3

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

Page 11: Intermediate Algebra Chapter 3

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

Page 12: Intermediate Algebra Chapter 3

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.

Page 13: Intermediate Algebra Chapter 3

Example

119 2 6

2x x

Page 14: Intermediate Algebra Chapter 3

Example

3 1x x

Page 15: Intermediate Algebra Chapter 3

Example

3 3x x

Page 16: Intermediate Algebra Chapter 3

Def: Equivalent Equations

• Equivalent equations are equations that have exactly the same solutions sets.

• Examples:

• 5 – 3x = 17

• -3x= 12

• x = -4

Page 17: Intermediate Algebra Chapter 3

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.

Page 18: Intermediate Algebra Chapter 3

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.

Page 19: Intermediate Algebra Chapter 3

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

Page 20: Intermediate Algebra Chapter 3

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

Page 21: Intermediate Algebra Chapter 3

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.”

Page 22: Intermediate Algebra Chapter 3

Useful Calculator Programs

• CIRCLE

• CIRCUM

• CONE

• CYLINDER

• PRISM

• PYRAMID

• TRAPEZOI

• APPS-AreaForm

Page 23: Intermediate Algebra Chapter 3

Robert Schuller – religious leader

• “Spectacular achievement is always preceded by spectacular preparation.”

Page 24: Intermediate Algebra Chapter 3

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

Page 25: Intermediate Algebra Chapter 3

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.”

Page 26: Intermediate Algebra Chapter 3

• Albert Einstein

»“In the middle of difficulty lies opportunity.”

Page 27: Intermediate Algebra Chapter 3

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.

Page 28: Intermediate Algebra Chapter 3

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

Page 29: Intermediate Algebra Chapter 3

Example solve by graphing

15 1

15 1

x x

x x

Page 30: Intermediate Algebra Chapter 3

Addition Property of Inequality

• If a < b, then a + c = b + c

• for all real numbers a, b, and c

Page 31: Intermediate Algebra Chapter 3

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

Page 32: Intermediate Algebra Chapter 3

Compound Inequalities 3.7

• Def: Compound Inequality: Two inequalities joined by “and” or “or”

Page 33: Intermediate Algebra Chapter 3

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

Page 34: Intermediate Algebra Chapter 3

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.

Page 35: Intermediate Algebra Chapter 3

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

Page 36: Intermediate Algebra Chapter 3

Solving inequalities involving “or”

• Solve each inequality in the compound inequality

• The solution set will be the union of the individual solution sets.

Page 37: Intermediate Algebra Chapter 3

Confucius

•“It is better to light one small candle than to curse the darkness.”

Page 38: Intermediate Algebra Chapter 3

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.

Page 39: Intermediate Algebra Chapter 3

Procedure for Absolute Value equation |ax+b|=c

• 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

Page 40: Intermediate Algebra Chapter 3

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

Page 41: Intermediate Algebra Chapter 3

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.

Page 42: Intermediate Algebra Chapter 3

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.

Page 43: Intermediate Algebra Chapter 3
Page 44: Intermediate Algebra Chapter 3

Joe Namath - quarterback

•“What I do is prepare myself until I know I can do what I have to do.”

Page 45: Intermediate Algebra Chapter 3

Intermediate Algebra 3.6

•Graphs

•Of

•Linear Inequalities

Page 46: Intermediate Algebra Chapter 3

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 >

Page 47: Intermediate Algebra Chapter 3

Def: Solution & solution setof 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.

Page 48: Intermediate Algebra Chapter 3

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.

Page 49: Intermediate Algebra Chapter 3

Linear inequalities on calculator

• Set =• Solve for Y• Input in Y=• Scroll left and scroll through icons

and press [ENTER]• Press [GRAPH]

Page 50: Intermediate Algebra Chapter 3

Calculator Problem

42

5y x

Page 51: Intermediate Algebra Chapter 3

Compound Inequalities

• Graph both inequalities

• AND – Intersection of both sets

• OR – Union of both sets.

Page 52: Intermediate Algebra Chapter 3

Abraham Lincoln U.S. President

•“Nothing valuable can be lost by taking time.”

Page 53: Intermediate Algebra Chapter 3