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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 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
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.
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 ( ) “The world looks like a multiplication table or a mathematical equation, which, turn it how you will, balances itself.”
Useful Calculator Programs CIRCLE CIRCUM CONE CYLINDER PRISM PYRAMID TRAPEZOI APPS-AreaForm
Robert Schuller – religious leader “Spectacular achievement is always preceded by spectacular preparation.”
Problem Solving 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
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 0, then ac < bc If a 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.
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.
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 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) 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 >
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 Set = Solve for Y Input in Y= Scroll left and scroll through icons and press [ENTER] Press [GRAPH]
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.”