EQUATIONS, INEQUALITIES & ABSOLUTE VALUE

2 CONTENT 2.1 Linear Equation 2.2 Quadratic Expression and Equations 2.3 Inequalities 2.4 Absolute value

2.1: Linear Equations

4 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

5 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

6 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.

7 Linear Equations A Linear Equation in one variable is equivalent to an equation of the form where a and b are real numbers and The linear equation has the single solution given by the formula Simplify the given equations first, to solve a linear equations

8 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)

9 Solve a Linear Equation Solve the following equations

10 Solve equations that lead to linear equations Solve the following equations

11 An equation with no solution Solve the following equations

12 Translating Written/Verbal Information into a Mathematical Model AdditionSubtractionMultiplicationDivisionEquals AndFromOfIntoIs PlusSubtractTimesOverEquals MoreLessProductDivided bySame as Added toFewerByQuotient ofMakes Together withMinusPercent ofRatio ofLeaves SumDifferenceMultiplied bya is to bYields TotalTake awayperEquivalent Increased byDecreased byResults in

13 Solve applied problems involving linear equations Example 1 A total of Rp 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?

2.2: Quadratic Expression &Equations

15 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

16 Quadratic Equations A Quadratic Equation in is an equation equivalent to one of the form where a, b and c are real numbers and A Quadratic Equation in the form is said to be in standard form 3 ways to solve quadratic equations a. Factoring b. Square root method c. Quadratic Formula

17 Solve a Quadratic Equation by Factoring Solve the following equations Repeated Solution / root of multiplicity 2 / double root When the left side, factors into 2 linear equations with same solution

18 Solve a Quadratic Equation by the Square Root Method Solve the following equations

19 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 Solve the following equations

20 Discriminant of a Quadratic Equation For a Quadratic Equation If there are two unequal real solutions If there is a repeated solution, a root of multiplicity 2 If there is no real solution (complex roots)

21 Examples Find a real solutions, if any, of the following equations

22 Application of Quadratic Equations Example 1 The quadratic function models the percentage of the U.S. population f (x), that was foreign-born x years after According to this model, in which year will 15% of the U.S. population be foreign-born?

2.3: Inequalities

24 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

25 Properties of Inequalities 1. If a < b and b < c then a < c 2. If a < b and c is any number, then a + c < b + c 3. If a < b and c is any number, then a – c < b – c 4. If a > 0 and b > 0 then a + b > 0 5. If a > 0 and b > 0 then ab > 0 6. If a 0 7. If a > b and –a < –b 8. If a –b 9. If a 0 then ac < bc 10. If a bc reciprocal property 13. reciprocal property

26 Solving Linear Inequalities Solve the following inequality and graph the solution set

27 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

28 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 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?

29 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. If the symbols are > or <, they are not solutions

30 Examples Solve the following inequality and graph the solution set

31 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 or, includes the values found in step 1 STEP 6: The solution set never includes the values found in Step 2 because they make the denominator equal to 0

32 Examples Solve the following inequality and graph the solution set

2.4: Absolute Value

34 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

35 What is Absolute Value The absolute value can be define as: The absolute value represents the distance of a point on the number line from the origin a - a

36 Properties of Absolute Value For any real number a and b

37 Properties of Absolute Value Equations involving absolute value Inequalities involving absolute value

38 Solve equations involving absolute value Solve the following equation

39 Solve inequalities involving absolute value Solve the following inequalities. Graph the solution set

40 Application of Absolute value The inequality 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

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