Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 1 Equations and Inequalities 1.1 Graphs and Graphing Utilities Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Plot points in the rectangular coordinate system. Graph equations in the rectangular coordinate system. Interpret information about a graphing utility’s viewing rectangle or table. Use a graph to determine intercepts. Interpret information given by graphs.

The Rectangular Coordinate System We draw a horizontal line and a vertical line that intersect at right angles. The horizontal line is the x-axis. The vertical line is the y-axis. The point of intersection for these axes is their zero points, known as the origin.

The Rectangular Coordinate System (continued) Positive numbers are shown to the right of the origin and above the origin. Negative numbers are shown to the left of the origin and below the origin.

Plotting Points in the Rectangular Coordinate System Each point in the rectangular coordinate system corresponds to an ordered pair of real numbers, (x, y). The first number in each pair, called the x-coordinate, denotes the distance and direction from the origin along the x-axis. The second number in each pair, called the y-coordinate, denotes the vertical distance and direction from the origin along the y-axis.

Example: Plotting Points in the Rectangular Coordinate System Plot the point (–2,4) To plot the point (–2,4), we move 2 units to the left of the origin and 4 units up.

Example: Plotting Points in the Rectangular Coordinate System (continued) Plot the point (4, –2) To plot the point (4, –2), we move 4 units to the right of the origin and 2 units down.

Graphs of Equations A relationship between two quantities can be expressed as an equation in two variables, such as A solution of an equation in two variables, x and y, is an ordered pair of real numbers with the following property: When the x-coordinate is substituted for x and the y-coordinate is substituted for y in the equation, we obtain a true statement.

Example: Graphing an Equation Using the Point-Plotting Method Select integers for x, starting with –4 and ending with 2.

Example: Graphing an Equation Using the Point-Plotting Method (continued) For each value of x, we find the corresponding value for y.

Example: Graphing an Equation Using the Point-Plotting Method (continued) We plot the points and connect them.

Graphing Utilities Graphing calculators and graphing software packages for computers are referred to as graphing utilities or graphers. To graph an equation in x and y using a graphing utility, enter the equation and specify the size of the viewing rectangle. The size of the viewing rectangle sets minimum and maximum values for both the x-axis and the y-axis. The [–10,10,1] by [–10,10,1] viewing rectangle is called the standard viewing rectangle.

Example: Understanding the Viewing Rectangle What is the meaning of a [–100,100,50] by [–100,100,10] viewing rectangle? The minimum x-value is –100. The maximum x-value is 100. The distance between consecutive tic marks on the x-axis is 50. The minimum y-value is – 100. The maximum y-value is 100. The distance between consecutive tic marks on the y-axis is 10.

Intercepts An x-intercept of a graph is the x-coordinate of a point where the graph intersects the x-axis. The y-coordinate corresponding to an x-intercept is always zero. A y-intercept of a graph is the y-coordinate of a point where the graph intersects the y-axis. The x-coordinate corresponding to a y-intercept is always zero.

Example: Identifying Intercepts Identify the x- and y-intercepts. The graph crosses the x-axis at (–3, 0). Thus, the x-intercept is –3. The graph crosses the y-axis at (0, 5). Thus, the y-intercept is 5.

Example: Interpret Information Given by Graphs Divorce rates are considerably higher for couples who marry in their teens. The equation models the percentage, d, of marriages that end in divorce after n years if the wife is under 18 at the time of marriage. Determine the percentage of marriages ending in divorce after 15 years when the wife is under 18 at the time of the marriage.

Example: Interpret Information Given by Graphs (continued) To determine the percentage of marriages ending in divorce after 15 years when the wife is under 18 at the time of the marriage we will use the equation where d is the percentage of marriages that end in divorce after n years. The percentage of marriages ending in divorce after 15 years when the wife is under 18 at the time of the marriage is 65%.

Example: Interpret Information Given by Graphs (continued) The graph of d = 4n + 5 is shown to the left. We calculated that 65% of marriages would end in divorce after 15 years. How can we check our answer with the graph? Our answer is the point (15, 65) which is on the graph.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 1 Equations and Inequalities 1.2 Linear Equations and Rational Equations Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19

Objectives: Solve linear equations in one variable. Solve linear equations containing fractions. Solve rational equations with variables in the denominators. Recognize identities, conditional equations, and inconsistent equations. Solve applied problems using mathematical models.

Definition of a Linear Equation A linear equation in one variable x is an equation that can be written in the form where a and b are real numbers, and

Generating Equivalent Equations An equation can be transformed into an equivalent equation by one or more of the following operations: 1. Simplify an expression by removing grouping symbols and combining like terms. 2. Add (or subtract) the same real number or variable expression on both sides of the equation. 3. Multiply (or divide) by the same nonzero quantity on both sides of the equation. 4. Interchange the two sides of the equation.

Solving a Linear Equation 1. Simplify the algebraic expression on each side by removing grouping symbols and combining like terms. 2. Collect all the variable terms on one side and all the numbers, or constant terms, on the other side. 3. Isolate the variable and solve. 4. Check the proposed solution in the original equation.

Example: Solving a Linear Equation Solve and check the linear equation: Check:

Example: Solving a Linear Equation Involving Fractions Solve and check: The LCD is 28, we will multiply both sides of the equation by 28.

Example: Solving a Linear Equation Involving Fractions (continued) Check:

Solving Rational Equations When we solved the equation we were solving a linear equation with constants in the denominators. A rational equation includes at least one variable in the denominator. For our next example, we will solve the rational equation

Example: Solving a Rational Equation We check for restrictions on the variable by setting each denominator equal to zero.

Example: Solving a Rational Equation (continued) The LCD is 18x, we will multiply both sides of the equation by 18x.

Example: Solving a Rational Equation to Determine When Two Equations are Equal Find all values of x for which y1 = y2 We begin by finding the restricted values. The restricted values are x = – 4 and x = 4

Example: Solving a Rational Equation to Determine When Two Equations are Equal (continued) We multiply both sides of the equation by the LCD

Example: Solving a Rational Equation to Determine When Two Equations are Equal (continued) 11 is not a restricted value. Therefore, x = 11.

Types of Equations: Identity, Conditional, Inconsistent An equation that is true for at least one real number is called a conditional equation. The examples that we have worked so far have been conditional equations. An equation that is true for all real numbers is called an identity. An equation that is not true for any real number is called an inconsistent equation.

Example: Categorizing an Equation This is a false statement. This equation is an example of an inconsistent equation. There is no solution.

Example: Categorizing an Equation This is a true statement. The solution set for this equation is the set of all real numbers. This equation is an identity.

Example: Solving Applied Problems Using Mathematical Models Persons with a low sense of humor have higher levels of depression in response to negative life events than those with a high sense of humor. This can be modeled by the formula In this formula, x represents the intensity of a negative life event and D is the average level of depression in response to that event. If the low humor group averages a level of depression of 10 in response to a negative life event, what is the intensity of that event?

Example: Solving Applied Problems Using Mathematical Models (continued) The intensity of the event is 3.7

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 1 Equations and Inequalities 1.3 Models and Applications Copyright © 2014, 2010, 2007 Pearson Education, Inc. 38

Objectives: Use linear equations to solve problems. Solve a formula for a variable.

Problem Solving with Linear Equations A model is a mathematical representation of a real-world situation. We will use the following general steps in solving word problems: Step 1 Let x represent one of the unknown quantities. Step 2 Represent other unknown quantities in terms of x. Step 3 Write an equation in x that models the conditions. Step 4 Solve the equation and answer the question. Step 5 Check the proposed solution in the original wording of the problem.

Example: Using Linear Equations to Solve Problems The median starting salary of a computer science major exceeds that of an education major by $21 thousand. The median starting salary of an economics major exceeds that of an education major by $14 thousand. Combined, their median starting salaries are $140 thousand. Determine the median starting salaries of education majors, computer science majors, and economics majors with bachelor’s degrees.

Example: Using Linear Equations to Solve Problems (continued) Step 1 Let x represent one of the unknown quantities. x = median starting salary of an education major Step 2 Represent other unknown quantities in terms of x. x + 21 = median starting salary of a computer science major x + 14 = median starting salary of an economics major Step 3 Write an equation in x that models the conditions.

Example: Using Linear Equations to Solve Problems (continued) Step 4 Solve the equation and answer the question. starting salary of an education major is x = 35 starting salary of a computer science major is x + 21 = 56 starting salary of an economics major is x + 14 = 49

Example: Using Linear Equations to Solve Problems (continued) Step 4 Solve the equation and answer the question. (continued) The median starting salary of an education major is $35 thousand, the median starting salary of a computer science major is $56 thousand, and the median starting salary of an economics major is $49 thousand.

Example: Using Linear Equations to Solve Problems (continued) Step 5 Check the proposed solution in the original wording of the problem. The problem states that combined, the median starting salaries are $140 thousand. Using the median salaries we determined in Step 4, the sum is $35 thousand + $56 thousand + $49 thousand, or $140 thousand, which verifies the problem’s conditions.

Example: Using Linear Equations to Solve Problems You are choosing between two texting plans. Plan A has a monthly fee of $15 with a charge of $0.08 per text. Plan B has a monthly fee of $3 with a charge of $0.12 per text. For how many text messages will the costs for the two plans be the same? Step 1 Let x represent one of the unknown quantities. x = charge per text Step 2 Represent other unknown quantities in terms of x. Plan A = 0.08x +15 Plan B = 0.12x + 3

Example: Using Linear Equations to Solve Problems (continued) Step 3 Write an equation in x that models the conditions. Step 4 Solve the equation and answer the question. The cost of the two plans will be the same for 300 text messages.

Example: Using Linear Equations to Solve Problems (continued) Step 5 Check the proposed solution in the original wording of the problem. The problem asks for how many text messages will the two plans be the same? The cost of Plan A for 300 text messages is 0.08(300) + 15 = 24 + 15 = 39 The cost of Plan B for 300 text messages is 0.12(300) + 3 = 36 + 3 = 39 For 300 text messages, the plan costs are equal, which verifies the problem’s conditions.

Example: Using Linear Equations to Solve Problems After a 30% price reduction, you purchase a new computer for $840. What was the computer’s price before the reduction? Step 1 Let x represent one of the unknown quantities. x = price before reduction Step 2 Represent other unknown quantities in terms of x. price of new computer = x – 0.3x Step 3 Write an equation in x that models the conditions.

Example: Using Linear Equations to Solve Problems (continued) Step 4 Solve the equation and answer the question. The price of the computer before the reduction was $1200. Step 5 Check the proposed solution in the original wording of the problem. The price before the reduction, $1200, minus the 30% reduction should equal the reduced price given in the original wording, $840.

Example: Using Linear Equations to Solve Problems (continued) Step 5 Check the proposed solution in the original wording of the problem. (continued) This verifies that the computer’s price before the reduction was $1200.

Example: Using Linear Equations to Solve Problems You inherited $5000 with the stipulation that for the first year the money had to be invested in two funds paying 9% and 11% annual interest. How much did you invest at each rate if the total interest earned for the year was $487? Step 1 Let x represent one of the unknown quantities. x = the amount invested at 9% Step 2 Represent other unknown quantities in terms of x. 5000 – x = the amount invested at 11% Step 3 Write an equation in x that models the conditions.

Example: Using Linear Equations to Solve Problems (continued) Step 4 Solve the equation and answer the question. the amount invested at 9% = x = 3150 the amount invested at 11% = 5000 – x = 1850 You should invest $3150 at 9% and $1850 at 11%.

Example: Using Linear Equations to Solve Problems (continued) Step 5 Check the proposed solution in the original wording of the problem. The problem states that the total interest from the dual investments should be $487. The interest earned at 9% is 0.09(3150) = 283.50 The interest earned at 11% is 0.11(1850) = 203.50 The total interest is 283.50 + 203.50 = 487. This verifies that the amount invested at 9% should be $3150 and that the amount invested at 11% should be $1850.

Example: Using Linear Equations to Solve Problems The length of a rectangular basketball court is 44 feet more than the width. If the perimeter of the basketball court is 288 feet, what are its dimensions? Step 1 Let x represent one of the unknown quantities. x = the width of the basketball court Step 2 Represent other unknown quantities in terms of x. x + 44 = the length of the basketball court Step 3 Write an equation in x that models the conditions.

Example: Using Linear Equations to Solve Problems (continued) Step 4 Solve the equation and answer the question. the width of the basketball court is x = 50 ft the length of the basketball court is x + 44 = 94 ft The dimensions of the basketball court are 50 ft by 94 ft

Example: Using Linear Equations to Solve Problems (continued) Step 5 Check the proposed solution in the original wording of the problem. The problem states that the perimeter of the basketball court is 288 feet. If the dimensions are 50 ft by 94 ft, then the perimeter is 2(50) + 2(94) = 100 + 188 = 288. This verifies the conditions of the problem.

Solving a Formula For A Variable Solving a formula for a variable means rewriting the formula so that the variable is isolated on one side of the equation.

Example: Solving a Formula for a Variable

Example: Solving a Formula for a Variable That Occurs Twice

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 1 Equations and Inequalities 1.4 Complex Numbers Copyright © 2014, 2010, 2007 Pearson Education, Inc. 61

Objectives: Add and subtract complex numbers. Multiply complex numbers. Divide complex numbers. Perform operations with square roots of negative numbers.

Complex Numbers and Imaginary Numbers The imaginary unit i is defined as The set of all numbers in the form a + bi with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The standard form of a complex number is a + bi.

Operations on Complex Numbers The form of a complex number a + bi is like the binomial a + bx. To add, subtract, and multiply complex numbers, we use the same methods that we use for binomials.

Example: Adding and Subtracting Complex Numbers Perform the indicated operations, writing the result in standard form:

Example: Multiplying Complex Numbers Find the product:

Conjugate of a Complex Number For the complex number a + bi, we define its complex conjugate to be a – bi. The product of a complex number and its conjugate is a real number.

Complex Number Division The goal of complex number division is to obtain a real number in the denominator. We multiply the numerator and denominator of a complex number quotient by the conjugate of the denominator to obtain this real number.

Example: Using Complex Conjugates to Divide Complex Numbers Divide and express the result in standard form: In standard form, the result is

Principal Square Root of a Negative Number For any positive real number b, the principal square root of the negative number – b is defined by

Example: Operations Involving Square Roots of Negative Numbers Perform the indicated operations and write the result in standard form:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 1 Equations and Inequalities 1.5 Quadratic Equations Copyright © 2014, 2010, 2007 Pearson Education, Inc. 72

Objectives: Solve quadratic equations by factoring. Solve quadratic equations by the square root property. Solve quadratic equations by completing the square. Solve quadratic equations using the quadratic formula. Use the discriminant to determine the number and type of solutions. Determine the most efficient method to use when solving a quadratic equation. Solve problems modeled by quadratic equations.

Definition of a Quadratic Equation A quadratic equation in x is an equation that can be written in the general form where a, b, and c are real numbers, with A quadratic equation in x is also called a second-degree polynomial equation in x.

The Zero-Product Principle To solve a quadratic equation by factoring, we apply the zero-product principle which states that: If the product of two algebraic expressions is zero, then at least one of the factors is equal to zero. If AB = 0, then A = 0 or B = 0.

Solving a Quadratic Equation by Factoring 1. If necessary, rewrite the equation in the general form , moving all nonzero terms to one side, thereby obtaining zero on the other side. 2. Factor completely. 3. Apply the zero-product principle, setting each factor containing a variable equal to zero. 4. Solve the equations in step 3. 5. Check the solutions in the original equation.

Example: Solving Quadratic Equations by Factoring Solve by factoring: Step 1 Move all nonzero terms to one side and obtain zero on the other side. Step 2 Factor

Example: Solving Quadratic Equations by Factoring (continued) Steps 3 and 4 Set each factor equal to zero and solve the resulting equations.

Example: Solving Quadratic Equations by Factoring (continued) Step 5 Check the solutions in the original equation. Check

Solving Quadratic Equations by the Square Root Property Quadratic equations of the form u2 = d, where u is an algebraic expression and d is a nonzero real number, can be solved by the Square Root Property: If u is an algebraic expression and d is a nonzero real number, then u2 = d has exactly two solutions: or Equivalently, If u2 = d, then

Example: Solving Quadratic Equations by the Square Root Property Solve by the square root property:

Completing the Square If x2 + bx is a binomial, then by adding , which is the square of half the coefficient of x, a perfect square trinomial will result. That is,

Example: Solving a Quadratic Equation by Completing the Square Solve by completing the square:

The Quadratic Formula The solutions of a quadratic equation in general form with , are given by the quadratic formula:

Example: Solving a Quadratic Equation Using the Quadratic Formula Solve using the quadratic formula: a = 2, b = 2, c = – 1

Example: Solving a Quadratic Equation Using the Quadratic Formula (continued)

The Discriminant We can find the solution for a quadratic equation of the form using the quadratic formula: The discriminant is the quantity which appears under the radical sign in the quadratic formula. The discriminant of the quadratic equation determines the number and type of solutions.

The Discriminant and the Kinds of Solutions to If the discriminant is positive, there will be two unequal real solutions. If the discriminant is zero, there is one real (repeated) solution. If the discriminant is negative, there are two imaginary solutions.

Example: Using the Discriminant Compute the discriminant, then determine the number and type of solutions: The discriminant, 81, is a positive number. There are two real solutions.

Example: Application The formula models a woman’s normal systolic blood pressure, P, at age A. Use this formula to find the age, to the nearest year, of a woman whose normal systolic blood pressure is 115 mm Hg. Solution: We will solve the equation

Example: Application (continued)

Example: Application (continued) The positive solution, indicates that 26 is the approximate age of a woman whose normal systolic blood pressure is 115 mm Hg.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 1 Equations and Inequalities 1.6 Other Types of Equations Copyright © 2014, 2010, 2007 Pearson Education, Inc. 93

Objectives: Solve polynomial equations by factoring. Solve radical equations. Solve equations with rational exponents. Solve equations that are quadratic in form. Solve equations involving absolute value. Solve problems modeled by equations.

Polynomial Equations A polynomial equation is the result of setting two polynomials equal to each other. The equation is in general form if one side is 0 and the polynomial on the other side is in descending powers of the variable. The degree of a polynomial equation is the same as the highest degree of any term in the equation.

Example: Solving a Polynomial Equation by Factoring Solve by factoring: Step 1 Move all nonzero terms to one side and obtain zero on the other side. Step 2 Factor.

Example: Solving a Polynomial Equation by Factoring (continued) Steps 3 and 4 Set each factor equal to zero and solve the resulting equations. The solution set is Step 5 Check the solutions in the original equation.

Radical Equations A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. We solve radical equations with nth roots by raising both sides of the equation to the nth power.

Solving Radical Equations Containing nth Roots 1. If necessary, arrange terms so that one radical is isolated on one side of the equation. 2. Raise both sides of the equation to the nth power to eliminate the isolated nth root. 3. Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and 2. 4. Check all proposed solutions in the original equation.

Example: Solving a Radical Equation Solve: Step 1 Isolate a radical on one side. Step 2 Raise both sides to the nth power.

Example: Solving a Radical Equation (continued) Step 3 Solve the resulting equation

Example: Solving a Radical Equation (continued) Step 4 Check the proposed solutions in the original equation. Check 6: Check 1: 1 is an extraneous solution. The only solution is x = 6.

Equations with Rational Exponents We know that rational exponents represent radicals: A radical equation with rational exponents can be solved by isolating the expression with the rational exponent, and raising both sides of the equation to a power that is the reciprocal of the rational exponent.

Example: Solving Equations Involving Rational Exponents Solve:

Equations That Are Quadratic in Form An equation that is quadratic in form is one that can be expressed as a quadratic equation using an appropriate substitution.

Example: Solving an Equation Quadratic in Form Solve: Notice that , we let u = x2

Equations Involving Absolute Value The absolute value of x describes the distance of x from zero on a number line. To solve an absolute value equation, we rewrite the absolute value equation without absolute value bars. If c is a positive real number and u represents an algebraic expression, then is equivalent to u = c or u = – c.

Example: Solving an Equation Involving Absolute Value Solve: The solution set is

Example: Applications The formula models weekly television viewing time, H, in hours, by annual income, I, in thousands of dollars. What annual income corresponds to 33.1 hours per week watching TV? An annual income of $225,000 corresponds to 33.1 hours per week watching TV.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 1 Equations and Inequalities 1.7 Linear Inequalities and Absolute Value Inequalities Copyright © 2014, 2010, 2007 Pearson Education, Inc. 110

Objectives: Use interval notation. Find intersections and unions of intervals. Solve linear inequalities. Recognize inequalities with no solution or all real numbers as solutions. Solve compound inequalities. Solve absolute value inequalities.

Solving an Inequality Solving an inequality is the process of finding the set of numbers that make the inequality a true statement. These numbers are called the solutions of the inequality and we say that they satisfy the inequality. The set of all solutions is called the solution set of the inequality.

Interval Notation The open interval (a,b) represents the set of real numbers between, but not including, a and b. The closed interval [a,b] represents the set of real numbers between, and including, a and b.

Interval Notation (continued) The infinite interval represents the set of real numbers that are greater than a. The infinite interval represents the set of real numbers that are less than or equal to b.

Parentheses and Brackets in Interval Notation Parentheses indicate endpoints that are not included in an interval. Square brackets indicate endpoints that are included in an interval. Parentheses are always used with or .

Example: Using Interval Notation Express the interval in set-builder notation and graph: [1, 3.5]

Finding Intersections and Unions of Two Intervals 1. Graph each interval on a number line. 2. a. To find the intersection, take the portion of the number line that the two graphs have in common. b. To find the union, take the portion of the number line representing the total collection of numbers in the two graphs.

Example: Finding Intersections and Unions of Intervals Use graphs to find the set: Graph of [1,3]: Graph of (2,6): Numbers in both [1,3] and (2,6): Thus,

Solving Linear Inequalities in One Variable A linear inequality in x can be written in one of the following forms : In general, when we multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol is reversed.

Example: Solving a Linear Inequality Solve and graph the solution set on a number line: The solution set is . The interval notation for this solution set is . The number line graph is:

Example: Solving Linear Inequalities [Recognize inequalities with no solution or all real numbers as solutions] Solve the inequality: The inequality is true for all values of x. The solution set is the set of all real numbers. In interval notation, the solution is In set-builder notation, the solution set is

Example: Solving a Compound Inequality Solve and graph the solution set on a number line: Our goal is to isolate x in the middle. In interval notation, the solution is [-1,4). In set-builder notation, the solution set is The number line graph looks like

Solving an Absolute Value Inequality If u is an algebraic expression and c is a positive number, 1. The solutions of are the numbers that satisfy 2. The solutions of are the numbers that satisfy or These rules are valid if is replaced by and is replaced by

Example: Solving an Absolute Value Inequality Solve and graph the solution set on a number line: We begin by expressing the inequality with the absolute value expression on the left side: We rewrite the inequality without absolute value bars. means or

Example: Solving an Absolute Value Inequality (continued) We solve these inequalities separately: The solution set is The number line graph looks like