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 • • • • • 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 • • • • • • • • 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 • • • • • • 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 • • • • Set = Solve for Y Input in Y= Scroll left and scroll through icons and press [ENTER] • Press [GRAPH] Calculator Problem 4 y x2 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.”