Chapter 2 Equations and Inequalities PDF

CHAPTER 2. EQUATIONS AND INEQUALITIES COLLEGE OF ENGINEERING Department of Civil Engineering  LINEAR EQUATIONS EQUATIONS – an algebraic equation in the variable x is a statement that two algebraic expressions in x are equal. Variable – refers to the unknown in the equation. Domain...

CHAPTER 2. EQUATIONS AND INEQUALITIES COLLEGE OF ENGINEERING Department of Civil Engineering  LINEAR EQUATIONS EQUATIONS – an algebraic equation in the variable x is a statement that two algebraic expressions in x are equal. Variable – refers to the unknown in the equation. Domain – set of numbers for which the algebraic expressions in the equation are defined. Solution – when the variable in an equation is replaced by a specific number, the resulting statement may either be true or false. If it is true, then that number is called solution (or root) of the equation Solution Set – set of all solutions A number that is a solution is said to satisfy the equation. COLLEGE OF ENGINEERING Department of Civil Engineering  LINEAR EQUATIONS IN ONE 𝑎𝑥VARIABLE +𝑏 =0 3 Classifications of a Linear Equation 1. Identity Equation – the solution set is all real numbers because any real number substituted for x will make the equation true. 2. Conditional Equation – is true for only some values of the variable. 3. Inconsistent Equation – results in a false statement. COLLEGE OF ENGINEERING Department of Civil Engineering SOLVING LINEAR EQUATIONS 1. 4x + 7 = 35 2. 4 x − 3 + 24 = 15 − 5(x + 6) COLLEGE OF ENGINEERING Department of Civil Engineering SOLVING A RATIONAL EQUATION 7 5 22 1. − = 2𝑥 3𝑥 3 2 1 2𝑥 2. − = 2 𝑥+1 𝑥−1 𝑥 −1 1 4 4𝑥+4 3. − = 2 2𝑥+5 2𝑥−1 4𝑥 +8𝑥−5 COLLEGE OF ENGINEERING Department of Civil Engineering LITERAL EQUATIONS 1. If p pesos is invested at the rate of 100r percent at simple interest for t years, and A pesos is the amount of the investment at t years, then the formula for determining A is 𝐴 = 𝑝(1 + 𝑟𝑡) Solve for p and t. COLLEGE OF ENGINEERING Department of Civil Engineering APPLICATIONS OF LINEAR EQUATION Solve a word problem: 1. Read the problem carefully so that you can understand it. 2. Determine the quantities that are known and those that are unknown. 3. Write down any numerical facts known about the variable. 4. Determine two algebraic expressions for the same number and form an equation from them. 5. Solve the equation you obtained. 6. Check the results. COLLEGE OF ENGINEERING Department of Civil Engineering Setting up a Linear Equation to Solve a Word Problem COLLEGE OF ENGINEERING Department of Civil Engineering Examples: The sum of two numbers is 9, and their difference is 6. What are the numbers? COLLEGE OF ENGINEERING Department of Civil Engineering Examples: Find two numbers whose sum is 7, given that one is 3 times the other. COLLEGE OF ENGINEERING Department of Civil Engineering Examples: If a rectangle has a length that is 3cm less than four times its width and its perimeter is 19cm, what are the dimensions? COLLEGE OF ENGINEERING Department of Civil Engineering Examples: A man invested part of Php 15,000 at 12% and the remainder at 8%. If his annual income from the two investments is Php 1,456.00, how much does he have invested at each rate? COLLEGE OF ENGINEERING Department of Civil Engineering Examples: A woman invested ₱25,000 in two business ventures. Last year, she made a profit of 15% from the first venture, but lost 5% from the second venture. If last year’s income from the two investments was equivalent to a return of 8% on the entire amount invested, how much had she invested in each venture? COLLEGE OF ENGINEERING Department of Civil Engineering Examples: Determine how many liters of a 7% acid solution and how many liters of a 12% acid solution should be mixed by a chemist to obtain 6L of a 10% acid solution? COLLEGE OF ENGINEERING Department of Civil Engineering Examples: Juan and Maria leave home at the same time in separate automobiles. Juan drives to his office, a distance of 24 km, and Maria drives to school, a distance of 28 km. They arrive at their destinations at the same time. What are their average rates, if the Juan’s average rate is 12 kph less than Maria? COLLEGE OF ENGINEERING Department of Civil Engineering Examples: One painter can paint a room in 12 hours and another can paint the same room in 10 hours. How long will it take to paint the room if they work together? COLLEGE OF ENGINEERING Department of Civil Engineering Examples: Find the angle between the hour hand and minute hand of a clock when the time is 4:20 COLLEGE OF ENGINEERING Department of Civil Engineering Examples: What time after 7:00 o’clock will the hands of a continuously driven clock are together? COLLEGE OF ENGINEERING Department of Civil Engineering QUADRATIC EQUATION  QUADRATIC EQUATION 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 When a quadratic equation is written in the above manner (with all nonzero terms on the left side and 0 on the right side), it is said to be in standard form. COLLEGE OF ENGINEERING Department of Civil Engineering Factoring 2 1. 𝑥 + 𝑥 − 6 = 0 2. Find the solution set: 2 5𝑥 2𝑥 1+ = 6 3 COLLEGE OF ENGINEERING Department of Civil Engineering Square Root Property Find the solution set: 2 1. 𝑥 − 25 = 0 2. 𝑥 2 − 11 = 0 COLLEGE OF ENGINEERING Department of Civil Engineering Completing the Square Solve the following quadratic equations by completing the square: 2 1. 𝑥 + 4𝑥 + 1 = 0 2. 3𝑥 2 − 2𝑥 − 6 = 0 COLLEGE OF ENGINEERING Department of Civil Engineering Quadratic Formula −𝑏 ± 𝑏 2 − 4𝑎𝑐 𝑥= 2𝑎 Find the solution set of the equations: 1. 6𝑥 2 = 10 + 11𝑥 2 5 2. 𝑥 + 𝑥 +1=0 3 COLLEGE OF ENGINEERING Department of Civil Engineering Discriminant of the Quadratic Equation 2 𝑏 − 4𝑎𝑐 It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect. COLLEGE OF ENGINEERING Department of Civil Engineering Discriminant of the Quadratic Equation Determine the character of the roots of each of the following equations. 2 1. 3𝑥 − 2𝑥 − 6 = 0 2. 4𝑥 2 − 12𝑥 + 9 = 0 COLLEGE OF ENGINEERING Department of Civil Engineering Applications of the Quadratic Equation A park contains a flower garden, 50m long and 30m wide, and a path of uniform width around it. If the area of the path is 600 sq.m., what is its width? COLLEGE OF ENGINEERING Department of Civil Engineering Applications of the Quadratic Equation It takes a boy 15 minutes longer to mow the lawn than it takes his sister, and if they both work together it takes them 56 minutes. How long does it take the boy to mow the lawn by himself? COLLEGE OF ENGINEERING Department of Civil Engineering OTHER EQUATIONS IN ONE VARIABLE One-Variable Equations involving Radicals Solve the equation: 2𝑥 + 5 + 𝑥 = 5 Find the solution set: 𝑥 𝑥−8=3 Find the solution set: 2𝑥 + 3 − 𝑥 − 2 − 2 = 0 COLLEGE OF ENGINEERING Department of Civil Engineering One-variable Equations that are Quadratic in Form 2 𝑎𝑢 + 𝑏𝑢 + 𝑐 = 0 where 𝑎 ≠ 0 and u is an algebraic expression x. COLLEGE OF ENGINEERING Department of Civil Engineering One-variable Equations that are Quadratic in Form Find the solution set of the given equation: 4 2 1. 𝑥 − 2𝑥 − 15 = 0 1 2. 2𝑥 2/3 − 5𝑥 − 3 = 0 3 1 2 1 3. 3 4𝑥 − − 4 4𝑥 − − 15 = 0 𝑥 𝑥 COLLEGE OF ENGINEERING Department of Civil Engineering One-Variable Equations of Higher Degree Find the solution set of the given equation: 3 1. 𝑥 =8 COLLEGE OF ENGINEERING Department of Civil Engineering Absolute Value Equation An absolute value equation in the form of 𝑎𝑥 + 𝑏 = 𝑐 has the following properties: 1. If 𝑐 < 0, 𝑎𝑥 + 𝑏 = 𝑐 has no solution. 2. If 𝑐 = 0, 𝑎𝑥 + 𝑏 = 𝑐 has one solution 3. If 𝑐 > 0, 𝑎𝑥 + 𝑏 = 𝑐 has two solutions. COLLEGE OF ENGINEERING Department of Civil Engineering Absolute Equation Solve the following absolute value equations: 1. 6𝑥 + 4 = 8 2. 3𝑥 + 4 = −9 COLLEGE OF ENGINEERING Department of Civil Engineering INEQUALITIES Inequalities 1. Trichotomy Property of Order states that either one point, a or b, on the real number line lies to the left of the other, or else they are the same point. That is, if a and b are real numbers, exactly one of the following three statements is true: a 0, then ac < bc 4. Multiplication Property: if a < b and c < 0, then ac > bc COLLEGE OF ENGINEERING Department of Civil Engineering Solving Inequalities in One Variable Algebraically Solve the following inequalities 1. 𝑥 − 15 < 4 2. 6 ≥ 𝑥 − 1 3. 𝑥 + 7 > 9 COLLEGE OF ENGINEERING Department of Civil Engineering Compound Inequalities A compound inequality includes two inequalities in one statement. There are two ways to solve compound inequalities: Separating them into two separate inequalities or leaving inequality intact and performing operations on all three parts at the same time COLLEGE OF ENGINEERING Department of Civil Engineering Compound Inequalities Solve the compound inequality 3 ≤ 2𝑥 + 2 < 6 COLLEGE OF ENGINEERING Department of Civil Engineering Absolute Value Inequalities For an algebraic expression X, and k > 0, an absolute value inequality is an inequality of the form 𝑋 < 𝑘 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 − 𝑘 < 𝑋 < 𝑘 𝑋 > 𝑘 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 𝑋 < −𝑘 𝑜𝑟 𝑋 > 𝑘 COLLEGE OF ENGINEERING Department of Civil Engineering Absolute Value Inequalities Find and show on the real number line the solution set of the inequality 2𝑥 − 7 < 9 COLLEGE OF ENGINEERING Department of Civil Engineering POLYNOMIAL, RATIONAL, AND ABSOLUTE VALUE INEQUALITIES Examples Solve the following inequality 1. 2 𝑥 − 8 < 2𝑥 2. 3 2 𝑥 − 2𝑥 − 3𝑥 ≤ 0 𝑥+4 3. 0, 𝑥 𝑏 𝑜𝑟 𝑥 < −𝑏 COLLEGE OF ENGINEERING Department of Civil Engineering Absolute Value Inequality Find the solution set of the inequality 2𝑥 − 7 < 9 COLLEGE OF ENGINEERING Department of Civil Engineering Get in Touch With Us Send us a message or visit us City of Batac, Ilocos Norte, Philippines (63) 77-600-0459 [email protected] Follow us for updates facebook.com/MMSUofficial www.mmsu.edu.ph