Chapter 1: Equations and Inequalities Section 1.1: Graphs and Graphing Utilities The Rectangular Coordinate System and Graphs Coordinate Geometry y II I P(a, b) (a, b) = (x, y) O x III IV A relationship between two quantities can be expressed as an equation in two variables; y = 4 x 2 An equation is a statement that two algebraic expressions are equal. A solution to an equation in two variables is an ordered pair (x, y) when substituted and evaluated yields a true numerical statement. A graph of an equation consists of the set of ordered pairs that are solutions to the equation. The graph of an equation in two variables is the visual representation of the solution set. From J. P. Wood, Spring 2012 1
Graph the equation: y = 4 x 2 (1) pick values of x (2) evaluate y values x y (3) plot ordered pairs (4) smoothly connect dots Now use TI-83 to graph To find intercepts: x-intercept : set y = 0 and solve for x y-intercept : set x = 0 and solve for y From J. P. Wood, Spring 2012 2
Section 1.2 Linear Equations and Rational Equations Definition of a Linear Equation: A linear equation in one variable x is an equation that can be written in the standard form: ax + b = 0 a,b are reals, a 0 A solution to a linear equation is a value of the variable when substituted and evaluated yields a true numerical statement. Solving equations centers around generating equivalent equations (rules on p 102) Solve the following: 3x 5 = 7 2(x 3) 17 = 13 3(x + 2) Use the TI-83 to verify the solution Linear Equations with Fractions: From J. P. Wood, Spring 2012 3
Rational Equations: A Rational Equation contains one or more rational expressions Solving a Rational Equation Solving a Rational Equation to determine when two equations are equal: Types of Equations: An equation that is true for all values in the domain of the variable is an identity: x 2 9 = (x + 3)(x 3) A conditional equation is one that is true for some but not all permissible values of the variable: 2x + 3 = 17 An inconsistent equation is one that is not true for any permissible value of the variable: x = x + 7 Determine the category (type) of the following equation: 2(x + 1) = 2x + 3 From J. P. Wood, Spring 2012 4
Modeling with Linear Equations Section 1.3: Mathematical Modeling: The process of translating phrases or sentences into algebraic expressions or equations. Strategy for Solving Word Problems: 1. Read problem carefully and then state in own words what the problem is looking for. Let x represent an unknown quantity. 2. If necessary write any other unknowns in terms of x. 3. Write an equation in x that models the verbal problem 4. Solve equation to answer the problem 5. Check solution in original wording of problem Example: You are choosing between two long-distance telephone plans. Plan A has a monthly fee of $20 with a charge of $0.05 per minute for all long-distance calls. Plan B has a monthly fee of $5 with a charge of $0.10 per minute for all long-distance calls. For how many minutes will the costs for the two plans be the same? Example: Your grandmother has $50,000 to invest and needs your help. Part of this money is to be invested in noninsured bonds paying 15% annual interest and the rest of the money is to be invested in a government insured CD paying 7% annual interest. She wants $6000 per year total interest earned from both of these investments. How much money should she invest in each? From J. P. Wood, Spring 2012 5
Example: A rectangular family room is three times as long as it is wide, and its perimeter is 60 ft. Find its dimensions. Common Formulas (p.123) For a circular cylinder, write the formula in terms of height. From J. P. Wood, Spring 2012 6
Section 1.4 Complex Numbers Let s solve: x 2 + 1 = 0 For the real numbers a and b, the number a + bi is a complex number a + bi = c + di if and only if a = c and b = d Operations with Complex Numbers Addition and Subtraction: Multiplication: Division using Complex Conjugates The Complex Conjugate of a + bi is a bi From J. P. Wood, Spring 2012 7
Roots of Negative Numbers Principal Square Root of a Negative Number b = i b Operations Involving Square Roots of Negative Numbers From J. P. Wood, Spring 2012 8
Section 1.5 Quadratic Equations A quadratic equation is an equation that can be written in the general form: ax 2 + bx + c = 0 Solving Quadratics by Factoring (rules on p.137) Remember the Zero Product Principle: x 2 7x +10 = 0 If ab = 0, then a=0 or b = 0 Solving Quadratics by the Square Root Property Extracting Square Roots: Let u 2 = d If u 2 = d then u =± d Solving Quadratics by Completing the Square: From J. P. Wood, Spring 2012 9
Solving Quadratics by using the Quadratic Formula For any ax 2 + bx + c = 0 x = 2 b b 4ac ± 2a The radicand of the Quadratic equation is called the discriminant ( Look at Table 1.2 on p.146) b 2 4ac If the discriminant is a perfect square, the quadratic is factorable 12x 2 10x 12 4 ways to solve a quadratic: 1. Factor (if factorable) 2. Extract the Root (if in u 2 = d format) 3. Complete the Square (always) 4. Quadratic Equation (always) Pythagorean Theorem a 2 + b 2 = c 2 c a b From J. P. Wood, Spring 2012 10
Section 1.6 Polynomial Equations Radical Equations Equations with Rational Exponents Solving form: If m is even m n x = k If m is odd Equations that are in Quadratic Form Equations Involving Absolute Value Absolute Value: If u = c then u = c or u = -c From J. P. Wood, Spring 2012 11
Section 1.7 Linear Inequalities and Absolute Value Inequalities Interval Notation: Open Interval (a, b) = {x a < x < b} Closed Interval [a, b] = {x a x b} Infinite Interval (a, ) = {x x > a} Infinite Interval (-, b] = {x x b} (More interval notations shown on p.174) Intersections and Unions of Intervals Find (1, 4) [2, 8] and (1, 4) U [2, 8] Solving Linear Inequalities in One Variable (Properties of Inequalities p 177) Addition Property of Inequality Positive Multiplication Property of Inequality From J. P. Wood, Spring 2012 12
Negative Multiplication Property of Inequality Solving a Linear Inequality Containing Fractions Inequalities with Unusual Solution Sets Solving Compound Inequalities Solving Inequalities with Absolute Value u < c iff -c < u < c u > c iff u < -c or u > c From J. P. Wood, Spring 2012 13