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Chapter 1 Equations and Inequalities 1.1 Linear Equations Definition 1 An equation in one variable is a statement in which two expressions, at least one containing the variable, are equal. Example 1 Solve the equation 3x 5 4. Definition 2 A linear equation in one variable is an equation that can be written in the form ax b 0 , where a and b are real numbers and a ≠ 0. Remark A linear equation is also called a first-degree equation. Solving Linear Equations in One Variable (1) Clear fractions: Multiply on both sides by the LCD to clear the equations of fractions if they occur. (2) Remove parentheses: Use the distributive property to remove parentheses if they occur. (3) Simplify: Simplify each side of the equation by combining like terms. (4) Isolate the variable: Get all variable terms on one side and all numbers on the other side by using the addition property of equality. (5) Solve: Get the variable alone by using the multiplication property of equality. Example 2 Solve the equation 3(2x – 4) = 7 – (x + 5). Example 3 Solve the equation 1 1 ( x 5) 4 (2 x 1). 2 3 1 Example 4 Solve the equation (2 y 1)( y 1) ( y 5)( 2 y 5). Example 5 Solve the equation 3 1 7 . x 2 x 1 ( x 1)( x 2) Example 6 Solve the equation 3x 3 2 . x 1 x 1 2 Solving for a Specified Variable All other letters are considered as numbers. Example 7 Solve PV = nRT for T. Example 8 Solve C 5 ( F 32) for F. 9 Steps for Setting Up Applied Problems (1) Read the problem carefully and identify what you are looking for. (2) Assign a letter (variable) to represent what you are looking for, and, if necessary, express any remaining unknown quantities in terms of this variable. (3) Make a list of all the known facts, and translate them into mathematical expressions. Set up the equation. (4) Solve the equation for the variable, and then answer the question, usually using a complete sentence. (5) Check the answer with the facts in the problem. Example 9 A total of $18,000 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? 3 Example 10 Shannon grossed $435 one week by working 52 hours. Her employer pays time-and-a-half for all hours worked in excess of 40 hours. With this information, can you determine Shannon’s regular hourly wage? 1.2Quadratic Equations Definition 1 A quadratic equation is an equation which can be written in the form ax 2 bx c 0 , where a, b and c are real numbers and a ≠ 0. Remark A quadratic equation written in the form of ax 2 bx c 0 is said to be in standard form. Remark A quadratic equation is also called a second-degree equation. Solving Quadratic Equations 1) Solving a Quadratic Equation by Factoring Zero-product Property If ab 0 , then a=0 or b=0. Example 1 Solve x 2 6 x 0. 4 Example 2 Solve 2 x 2 x 3. Remark When the left side factors into two linear equations with the same solution, the quadratic equation is said to have a repeated solution. We also call this solution a root of multiplicity 2, or a double root. Example 3 Solve x 2 6 x 9 0. 2) The Square Root Method If x 2 p and p 0 , then x Example 4 Solve (a) x 2 5 p or x p . (b) ( x 2) 2 16 3) Complete square Procedure for completing a square Start Add m x 2 mx ( )2 2 Result m m x 2 mx ( ) 2 ( x ) 2 2 2 5 Example 5 Determine the number that must be added to each expression to complete the square. Then factor the resulted expression. (1) x 2 8 x (2) x 2 20 x Example 6 Solve by completing the square. (1) x 2 5 x 4 0 (2) 2 x 2 8 x 5 0 4) The Quadratic Formula For ax 2 bx c 0 , a 0 , we have the quadratic formula b 2 4ac (1) If (2) If (3) If b b 2 4ac . x 2a is called the discriminant of the quadratic equation. b 2 4ac 0 , there are two unequal real solutions. b 2 4ac 0 , there is a repeated solution, a root of multiplicity 2. b 2 4ac 0 , there is no solution. 6 Example 7 Solve 3x 2 5 x 1 0 Example 8 Solve 25 2 x 30 x 18 0. 2 Example 9 Solve 3x 2 2 4 x 7 Example 10 Solve 9 3 2 0, x 0 x x2 1.4 Radical Equations; Equations Quadratic in Form; Factorable Equations Definition 1 When the variable in an equation occurs in a radical (square root, cube root, and so on), the equation is called a radical equation. To solve a radical equation, (1) Isolate the most complicated radical on one side of the equation. (2) Eliminate it by raising each side to a power equal to the index of the radical. (3) We need to check all answers when working with radical equations. Example 1 Solve 3 2 x 4 2 0. 8 Example 2 Solve x 1 x 7. Example 3 Solve 2 x 3 x 2 2. Definition 2 If an appropriate substitution u transforms an equation in the form au 2 bu c 0, u ≠ 0 then the original equation is called an equation quadratic in form. 9 Example 4 Solve ( x 2) 2 11( x 2) 12 0. Example 5 Solve x 6 9 x 3 8 0. 10 Example 6 Solve 3 2 x x. Factorable Equations Example 7 Solve x 4 4 x 2 . Example 8 Solve x 3 x 2 4 x 4 0. 11 1.5 Solving Inequalities Intervals INTERVAL Open interval (a, b) Closed interval [a, b] Half-open interval [a, b) Half-open interval (a, b] Interval [a, ) Interval (a, ) Interval (, a] Interval ( , a ) Interval (, ) INEQUALITY GRAPH Example 1 Write each inequality using interval notation. (a) 1 x 3 , (b) 4 x 0 , (c) x 5 , (d) x 1 Example 2 Write each interval as an inequality involving x (a) [1,4), (b) (2, ), (c) [2,3], (d) (- ,3 ) Properties of Inequalities Nonnegative Property: a 2 0 . Addition Property for Inequalities: If a < b, then a + c < b + c. If a > b, then a + c > b + c. Multiplication Properties for Inequalities: If a < b and if c > 0, then ac < bc. If a > b and if c > 0, then ac > bc. If a < b and if c < 0, then ac > bc. If a > b and if c < 0, then ac < bc. 12 Reciprocal Properties: 1 0. a 1 If a 0 , then 0 . a If a 0 , then Example 3 (a) If x 5 , then (b) If x 2 , then Example 4 (a) If 2x 6 , then (b) If x 12 , then 3 (c) If 4x 8 , then Solving Inequalities Example 5 Solve the inequality 3 2x 5 . Example 6 Solve the inequality 4x 7 2x 3 . Solving Combined Inequalities (1) Keep the variable in the middle. (2) Work with all three expressions at the same time. 13 Example 7 Solve the inequality 5 3x 2 1 . Example 8 Solve the inequality 1 3 5x 9. 2 Example 9 Solve the inequality (4 x 1) 1 0 . Example 10 If 1 x 4 , find a and b so that a 2x 1 b . 14 1.6 Equations and Inequalities Involving Absolute Value Equations Involving Absolute Value If a is a positive real number and if u is an algebraic expression, then |u| = a is equivalent to u = a, or u = -a. Example 1 Solve the equation x 4 13 . Example 2 Solve the equation 2 x 3 2 7. Example 3 Solve the equation x 2 x 1 1 . Inequalities Involving Absolute Value (I) If a is a positive real number and if u is an algebraic expression, then |u| < a is equivalent to -a <u < a |u| a is equivalent to -a u a 15 Example 4 Solve the inequality 2 x 4 3 . Example 5 Solve the inequality 1 4 x 1 6 . Inequalities Involving Absolute Value (II) If a is a positive real number and if u is an algebraic expression, then |u| > a is equivalent to u < -a or u > a |u| ≥ a is equivalent to u -a or u ≥ a Example 6 Solve the inequality 2 x 5 3 . Example 7 Solve the inequality x 3 1 1 . 16 1.7 Applications: Interest, Mixture, Uniform Motion, Constant Rate Jobs 1) Solve Interest Problems Simple Interest Formula If a principal of P dollars is borrowed for a period of t years at a per annum interest rate r, expressed as a decimal, the interest I charged is I = Prt. Example 1 Candy has $70,000 to invest and requires an overall rate of return of 9%. She can invest in a safe, government-insured certificate of deposit, but it only pays 8%. To obtain 9%, she agrees to invest some of her money in noninsured corporate bonds paying 12%. How much should be placed in each investment to achieve her goal? 2) Solve Mixture Problems Example 2 The manager of a Starbucks store decides to experiment with a new blend of coffee. She will mix some B grade coffee that sells for $5 per pound with some A grade coffee that sells for $10 per pound to get 100 pounds of the new blend. The selling price of the new blend is to be $7 per pound, and there is no difference in revenue from selling the new blend versus selling the other types. How many pounds of the B grade and A grade coffees are required? 17 3) Solve Uniform Motion Problems Uniform Motion Formula If an object moves at an average velocity v, the distance s covered in time t is given by S = vt. That is, Distance = Velocity ∙ Time Example 3 Tanya, who is a long-distance runner, runs at an average velocity of 8 miles per hour (mi/hr). Two hours after Tanya leaves your house, you leave in your Honda and follow the same route. If your average velocity is 40mi/hr, how long will it be before you catch up to Tanya? How far will each of you be from your home? 4) Solve Constant Rate Job Problems If a job can be done in t units of time, 1 of the job is done in 1 unit of time. t Example 4 One computer can do a job twice as fast as another. Working together, both computers can do the job in 2 hours. How long would it take each computer, working alone, to do the job? 18