Download - Intermediate Algebra Chapter 3
![Page 1: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/1.jpg)
Intermediate Algebra Chapter 3
•Linear Equations
•and
•Inequalities
![Page 2: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/2.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/3.jpg)
Intermediate Algebra 3.1
•Introduction
•To
•Linear Equations
![Page 4: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/4.jpg)
Def: Equation
•An equation is a statement that two algebraic expressions
have the same value.
![Page 5: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/5.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/6.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/7.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/8.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/9.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/10.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/11.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/12.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/13.jpg)
Example
119 2 6
2x x
![Page 14: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/14.jpg)
Example
3 1x x
![Page 15: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/15.jpg)
Example
3 3x x
![Page 16: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/16.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/17.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/18.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/19.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/20.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/21.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/22.jpg)
Useful Calculator Programs
• CIRCLE
• CIRCUM
• CONE
• CYLINDER
• PRISM
• PYRAMID
• TRAPEZOI
• APPS-AreaForm
![Page 23: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/23.jpg)
Robert Schuller – religious leader
• “Spectacular achievement is always preceded by spectacular preparation.”
![Page 24: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/24.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/25.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/26.jpg)
• Albert Einstein
»“In the middle of difficulty lies opportunity.”
![Page 27: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/27.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/28.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/29.jpg)
Example solve by graphing
15 1
15 1
x x
x x
![Page 30: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/30.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/31.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/32.jpg)
Compound Inequalities 3.7
• Def: Compound Inequality: Two inequalities joined by “and” or “or”
![Page 33: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/33.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/34.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/35.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/36.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/37.jpg)
Confucius
•“It is better to light one small candle than to curse the darkness.”
![Page 38: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/38.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/39.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/40.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/41.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/42.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/43.jpg)
![Page 44: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/44.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/45.jpg)
Intermediate Algebra 3.6
•Graphs
•Of
•Linear Inequalities
![Page 46: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/46.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/47.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/48.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/49.jpg)
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](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/50.jpg)
Calculator Problem
42
5y x
![Page 51: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/51.jpg)
Compound Inequalities
• Graph both inequalities
• AND – Intersection of both sets
• OR – Union of both sets.
![Page 52: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/52.jpg)
Abraham Lincoln U.S. President
•“Nothing valuable can be lost by taking time.”
![Page 53: Intermediate Algebra Chapter 3](https://reader033.vdocuments.site/reader033/viewer/2022061614/56813652550346895d9dd834/html5/thumbnails/53.jpg)