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EQUATIONS, INEQUALITIES & ABSOLUTE VALUE CONTENT 2.1 Linear Equation 2.2 Quadratic Expression and Equations 2.3 Inequalities 2.4 Absolute value 2 2.1: Linear Equations Objectives • At the end of this topic, you should be able to • • • • Define linear equations Solve a linear equation Solve equations that lead to linear equations Solve applied problems involving linear equations 4 Equation in one variable A statements in which 2 expressions (sides) at least one containing the variable are equal It may be TRUE or FALSE depending on the value of the variable. The admissible values of the variable (those in the domain of the variable), if any, that result in a TRUE statement are called solutions or root. To solve an equation means to find all the solutions of the equation 5 Equation in one variable, cont… An equation will have only one solution or more than one solution or no real solutions or no solution Solution set – the set of solutions of an equation, {a} Identity – An equation that is satisfied for every value of the variable for which both sides are defined Equivalent equations – Two or more equations that have the same solution set. 6 Linear Equations A Linear Equation in one variable is equivalent to an equation of the form ax b 0 where a and b are real numbers and a 0 The linear equation has the single solution given by the formula x b a Simplify the given equations first, to solve a linear equations 7 Steps for Solving a Linear Equation STEP 1: If necessary, clear the equation of fractions by multiplying both sides by the least common multiple (LCM) of the denominators of all the fractions. STEP 2: Remove all parentheses and simplify STEP 3: Collect all terms containing the variable on one side and all remaining terms on the other side. STEP 4: Check your solution (s) 8 Solve a Linear Equation Solve the following equations 1. 3x 4 x 2. 2t 6 3 t 1 1 3. x 5 4 2 x 1 2 3 3 1 1 4. y2 y 2 2 2 9 Solve equations that lead to linear equations Solve the following equations 1. 5 3 x2 x 1 4 5 2. 5 y 2y 3. 3 1 7 x2 x 1 x 1 x 2 4. 2 y 1 y 1 y 5 2 y 5 10 An equation with no solution Solve the following equations 1. 3x 3 2 x 1 x 1 3x 1 2 3x 2. x2 2 x 11 Translating Written/Verbal Information into a Mathematical Model Addition Subtraction Multiplication Division Equals And From Of Into Is Plus Subtract Times Over Equals More Less Product Divided by Same as Added to Fewer By Quotient of Makes Together with Minus Percent of Ratio of Leaves Sum Difference Multiplied by a is to b Yields Total Take away per Increased by Decreased by Equivalent Results in 12 Solve applied problems involving linear equations Example 1 A total of Rp.45.000.000 is invested, some in stocks and some in bonds. If the amount invested in bonds is half that invested in stocks, how much is invested in each category? 13 2.2: Quadratic Expression &Equations Objectives • At the end of this topic you should be able to • Define quadratic expressions and equations • Solve quadratic equations by factorization, square root method, and quadratic formula • Recognize the types of roots of a quadratic equation based on the value of discriminant • Solve applied problems involving quadratic equations 15 Quadratic Equations A Quadratic Equation in is an equation equivalent to one of the form ax 2 bx c 0 where a, b and c are real numbers and a 0 A Quadratic Equation in the form ax 2 bx c 0 is said to be in standard form 3 ways to solve quadratic equations a. Factoring b. Square root method c. Quadratic Formula 16 Solve a Quadratic Equation by Factoring Solve the following equations 1. x 2 5 x 6 0 3. 9 x 2 6 x 1 0 2. 2x 2 x 3 4. 3x 2 5 x 2 0 Repeated Solution / root of multiplicity 2 / double root When the left side, factors into 2 linear equations with same solution 17 Solve a Quadratic Equation by the Square Root Method If x p and p 0, then x p 2 Solve the following equations 1. x 5 2 2. x 2 2 16 18 Solve a Quadratic Equation by the Quadratic Formula Use the method of completing the square to obtain a general formula for solving the quadratic equation From ax 2 bx c 0 to b b2 4ac x 2a Solve the following equations 1. x 2 6 x 16 0 2. x 2 5 x 4 0 3. 2 x 2 8 x 5 0 4. 2x 2 3x 24 0 19 Discriminant of a Quadratic Equation For a Quadratic Equation b 2 4ac 0 If there are two unequal real solutions b 2 4ac 0 If ax 2 bx c 0 there is a repeated solution, a root of multiplicity 2 b 2 4ac 0 If there is no real solution (complex roots) 20 Examples Find a real solutions, if any, of the following equations 1. 3 x 2 5 x 1 0 3. 3 x 2 4 x 2 0 25 2 2. x 30 x 18 0 2 3 2 4. 9 2 0 x x 5. x 2 x 1 0 6. x 1 x 7 21 Application of Quadratic Equations Example 1 2 f x 0.0049 x 0.361x 11.79 The quadratic function models the percentage of the U.S. population f (x), that was foreign-born x years after 1930. According to this model, in which year will 15% of the U.S. population be foreign-born? 22 2.3: Inequalities Objectives • At the end of this topic you should be able to • • • • Relate the properties of inequalities Define and Solve linear inequalities Define Solve quadratic inequalities Understand and solve rational inequalities involving linear and quadratic expression 24 Properties of Inequalities 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. If a < b and b < c then a < c If a < b and c is any number, then a + c < b + c If a < b and c is any number, then a – c < b – c If a > 0 and b > 0 then a + b > 0 If a > 0 and b > 0 then ab > 0 If a < b then b – a > 0 If a > b and –a < –b If a < b and –a > –b If a < b and c > 0 then ac < bc If a < b and c < 0 then ac > bc a2 0 1 0 12. a 13. If a 0 then 1 0 [email protected] a If a 0 then reciprocal property reciprocal property 25 Solving Linear Inequalities Solve the following inequality and graph the solution set 1. 3 2 x 5 2. 4x 7 2x 3 3. 1 1 9 x 5 2 x x 1 3 4 [email protected] 4. 5 3 x 2 1 3 5x 5. 1 9 2 6. 4 x 1 1 0 26 Solves problems involving linear inequalities At least, minimum of, no less than At most, maximum of, no more than Is greater than, more than Is less than, smaller than 27 Examples Sasha’s grade in her math course is calculated by the average of four tests. To receive an A for this course, she needs an average at least 89.5. If her current test scores are 84, 92, and 94, what range of scores can she make on the last test to receive an A for the course? A painter charges RM80 plus RM1.50 per square foot. If a family is willing to spend no more than RM500, then what is the range of square footage they can afford? 28 Solving Quadratic inequalities Step 1 - solve the related quadratic equation Step 2 – plot the solution on a number line Step 3 – Choose a test number from each interval & substitute the number into the inequality If the test number makes the inequality true All numbers in that interval will solve the inequality If the test number makes the inequality false No numbers in that interval will solve the inequality Step 4 – State the solution set of the inequality ( It is a union of all intervals that solves the inequality) If the inequality symbols are or , then the values from Step 2 are included. 29 [email protected] If the symbols are > or <, they are not solutions Examples Solve the following inequality and graph the solution set 1. x x 6 0 2 2. x 3x 0 2 3. x 1 4. x 1 2 2 2 2 30 Solving rational inequality STEP 1: Solve the related equation STEP 2: Find all values that make any denominator equal to 0 STEP 3: Plot the number found in Step 1 and 2 on a number line STEP 4: Choose a test number from each interval and determine whether it solves the inequality. STEP 5: The solution set is the union of all regions whose test number solves the inequality. If the inequality symbol is includes the values found in step 1 or , STEP 6: The solution set never includes the values found in Step 2 because they make the denominator equal to 0 [email protected] 31 Examples Solve the following inequality and graph the solution set 1. 2. x4 0 x2 x5 4 x 1 3. 1 2 x 1 x 1 4. x2 3 x4 32 2.4: Absolute Value Objectives • At the end of this topic you should be able to • Define absolute value • Understand, state and use the properties of absolute value • Solve problems on equations and inequalities involving absolute value 34 What is Absolute Value The absolute value can be define as: a, a 0 a a, a 0 The absolute value represents the distance of a point on the number line from the origin a -a 35 Properties of Absolute Value For any real number a and b 1. a 0 2. a a 3. ab ba 4. a b ba 5. ab a b 6. a a b b ,b 0 36 Properties of Absolute Value Equations involving absolute value If a is a positive real number and if y is any algebraic expression, then y a is equivalent to y a or y a Inequalities involving absolute value If a is a positive real number and if y is any algebraic expression, then y a is equivalent to a y a y a is equivalent to a y a In other words, y a is equivalent to a y and y a y a is equivalent to y a and y a [email protected] u a is equivalent to u a and u a 37 Solve equations involving absolute value Solve the following equation 1. x 4 13 1 3. 5 x =1 2 2. 3 x9 4. x 16 =0 2 38 Solve inequalities involving absolute value Solve the following inequalities. Graph the solution set 1. 2x 4 3 3. x 3 2. 1 4 x <5 4. 2 x 5 3 39 Application of Absolute value The inequality x 9 2.9 describes the percentage of children in the population who think that being grounded is a bad thing about being kid. Solve the inequality and interpret the solution 40 Thank You