College Algebra Exam 2 Material

Quadratic Applications Application problems may give rise to all types of equations, linear, quadratic and others Here we take a look at two that lead to quadratic equations

Example Two boys have two way radios with a range of 5 miles, how long can they communicate if they leave from the same point at the same time with one traveling north at 10 mph and the other traveling east at 7 mph? D = R T N boy E boy

Example A rectangular piece of metal is 2 inches longer than it is wide. Four inch squares are cut from each corner to make a box with a volume of 32 cubic inches. What were the original dimensions of the metal? Unknowns L Rec W Rec L Box W Box .

Homework Problems Section: 1.5 Page: 130 Problems: 5 – 9, 21 – 22 MyMathLab Assignment 22 for practice

Other Types of Equations Thus far techniques have been discussed for solving all linear and quadratic equations and some higher degree equations Now address techniques for identifying and solving many other types of equations

Solving Higher Degree Polynomial Equations So far methods have been discussed for solving first and second degree polynomial equations Higher degree polynomial equations may sometimes be solved using the “zero factor method” or, the “zero factor method” in combination with the “quadratic formula” or the “square root property” Consider two examples:

Example One Make one side zero: Factor non-zero side: Apply zero factor property and solve:

Example Two x3 + x2 – 9x – 9 = 0 One side is already zero, so factor non-zero side x2(x + 1) – 9(x + 1) = 0 (x + 1)(x2 – 9) = 0 Apply zero factor property and solve: x + 1 = 0 OR x2 – 9 = 0 x = -1 x2 = 9 x = ± 3

Homework Problems Section: 1.4 Page: 124 Problems: 59 – 62 MyMathLab Assignment 23 for practice

Rational Equations Technical Definition: An equation that contains a rational expression Practical Definition: An equation that has a variable in a denominator Example:

Solving Rational Equations Find “restricted values” for the equation by setting every denominator that contains a variable equal to zero and solving Find the LCD of all the fractions and multiply both sides of equation by the LCD to eliminate fractions Solve the resulting equation to find apparent solutions Solutions are all apparent solutions that are not restricted

Example

Homework Problems Section: 1.6 Page: 144 Problems: Odd: 1 – 25 MyMathLab Assignment 24 for practice

“Quadratic in Form” Equation An equation is “quadratic in form” if the same algebraic expression is found twice where one time the exponent on the expression is twice as big as it is the other time Examples: m6 – 7m3 – 8 = 0 8(x – 4)4 – 10(x – 4)2 + 3 = 0

Solving Equations that are Quadratic in Form Make a substitution by letting “u” equal the repeated expression with exponent that is half of the other Solve the resulting quadratic equation for “u” Make a reverse substitution for “u” Solve the resulting equation

Example of Solving an Equation that is Quadratic in Form

Example of Solving an Equation that is Quadratic in Form

Homework Problems Section: 1.6 Page: 145 Problems: All: 61 – 64, 73 – 74 MyMathLab Assignment 25 for practice

“Negative Integer Exponent” Equation An equation is a “negative integer exponent equation” if it has a variable expression with a negative integer exponent Examples:

Solving “Negative Integer Exponent” Equations If the equation is “quadratic in form”, begin solution by that method Otherwise, use the definition of negative exponent to convert the equation to a rational equation and solve by that method

Example of Solving Equation With Negative Integer Exponents

Example of Solving Equation With Negative Integer Exponents

Homework Problems Section: 1.6 Page: 145 Problems: 75, 76 MyMathLab Assignment 26 for practice MyMathLab Homework Quiz 5/6 will be due for a grade on the date of our next class meeting

Radical Equations An equation is called a radical equation if it contains a variable in a radicand Examples:

Solving Radical Equations Isolate ONE radical on one side of the equal sign Raise both sides of equation to power necessary to eliminate the isolated radical Solve the resulting equation to find “apparent solutions” Apparent solutions will be actual solutions if both sides of equation were raised to an odd power, BUT if both sides of equation were raised to an even power, apparent solutions MUST be checked to see if they are actual solutions

Why Check When Both Sides are Raised to an Even Power? Raising both sides of an equation to a power does not always result in equivalent equations If both sides of equation are raised to an odd power, then resulting equations are equivalent If both sides of equation are raised to an even power, then resulting equations are not equivalent (“extraneous solutions” may be introduced) Raising both sides to an even power, may make a false statement true: Raising both sides to an odd power never makes a false statement true: .

Example of Solving Radical Equation

Example of Solving Radical Equation

Example of Solving Radical Equation

Homework Problems Section: 1.6 Page: 144 Problems: Odd: 27 – 51, 55 – 57 MyMathLab Assignment 27 for practice

Rational Exponent Equations An equation in which a variable expression is raised to a “fractional power” Example:

Solving Rational Exponent Equations If the equation is quadratic in form, solve that way Otherwise, solve essentially like radical equations Isolate ONE rational exponent expression Raise both sides of equation to power necessary to change the fractional exponent into an integer exponent Solve the resulting equation to find “apparent solutions” Apparent solutions will be actual solutions if both sides of equation were raised to an odd power, but if both sides of equation were raised to an even power, apparent solutions MUST be checked to see if they are actual solutions

Example

Homework Problems Section: 1.6 Page: 145 Problems: All: 53 – 54, 59 – 60, 65 – 72 MyMathLab Assignment 28 for practice

Definition of Absolute Value “Absolute value” means “distance away from zero” on a number line Distance is always positive or zero Absolute value is indicated by placing vertical parallel bars on either side of a number or expression Examples: The distance away from zero of -3 is shown as: The distance away from zero of 3 is shown as: The distance away from zero of u is shown as:

Absolute Value Equation An equation that has a variable contained within absolute value symbols Examples: | 2x – 3 | + 6 = 11 | x – 8 | – | 7x + 4 | = 0 | 3x | + 4 = 0

Solving Absolute Value Equations Isolate one absolute value that contains an algebraic expression, | u | If the other side is a negative number there is no solution (distance can’t be negative) If the other side is zero, then write: u = 0 and Solve If the other side is “positive n”, then write: u = n OR u = - n and Solve If the other side is another absolute value expression, | v |, then write: u = v OR u = - v and Solve

Example of Solving Absolute Value Equation

Example of Solving Absolute Value Equation

Example of Solving Absolute Value Equation

Homework Problems Section: 1.8 Page: 164 Problems: Odd: 9 – 23, 41 – 43, 67 – 69 MyMathLab Assignment 29 for practice MyMathLab Homework Quiz 7 will be due for a grade on the date of our next class meeting

Inequalities An equation is a comparison that says two algebraic expressions are equal An inequality is a comparison between two or three algebraic expressions using symbols for: greater than: greater than or equal to: less than: less than or equal to: Examples: .

Inequalities There are lots of different types of inequalities, and each is solved in a special way Inequalities are called equivalent if they have exactly the same solutions Equivalent inequalities are obtained by using “properties of inequalities”

Properties of Inequalities Adding or subtracting the same number to all parts of an inequality gives an equivalent inequality with the same sense (direction) of the inequality symbol Multiplying or dividing all parts of an inequality by the same POSITIVE number gives an equivalent inequality with the same sense (direction) of the inequality symbol Multiplying or dividing all parts of an inequality by the same NEGATIVE number and changing the sense (direction) of the inequality symbol gives an equivalent inequality

Solutions to Inequalities Whereas solutions to equations are usually sets of individual numbers, solutions to inequalities are typically intervals of numbers Example: Solution to x = 3 is {3} Solution to x < 3 is every real number that is less than three Solutions to inequalities may be expressed in: Standard Notation Graphical Notation Interval Notation

Two Part Linear Inequalities A two part linear inequality is one that looks the same as a linear equation except that an equal sign is replaced by inequality symbol (greater than, greater than or equal to, less than, or less than or equal to) Example:

Expressing Solutions to Two Part Inequalities “Standard notation” - variable appears alone on left side of inequality symbol, and a number appears alone on right side: “Graphical notation” - solutions are shaded on a number line using arrows to indicate all numbers to left or right of where shading ends, and using a parenthesis to indicate that a number is not included, and a square bracket to indicate that a number is included “Interval notation” - solutions are indicated by listing in order the smallest and largest numbers that are in the solution interval, separated by comma, enclosed within parenthesis and/or square bracket. If there is no limit in the negative direction, “negative infinity symbol” is used, and if there is no limit in the positive direction, a “positive infinity symbol” is used. When infinity symbols are used, they are always used with a parenthesis.

Solving Two Part Linear Inequalities Solve exactly like linear equations EXCEPT: Always isolate variable on left side of inequality Correctly apply principles of inequalities (In particular, always remember to reverse sense of inequality when multiplying or dividing by a negative)

Example of Solving Two Part Linear Inequalities

Homework Problems Section: 1.7 Page: 156 Problems: Odd: 13 – 23 MyMathLab Assignment 30 for practice

Three Part Linear Inequalities Consist of three algebraic expressions compared with two inequality symbols Both inequality symbols MUST have the same sense (point the same direction) AND must make a true statement when the middle expression is ignored Good Example: Not Legitimate: .

Expressing Solutions to Three Part Inequalities “Standard notation” - variable appears alone in the middle part of the three expressions being compared with two inequality symbols: “Graphical notation” – same as with two part inequalities: “Interval notation” – same as with two part inequalities:

Solving Three Part Linear Inequalities Solved exactly like two part linear inequalities except that: solution is achieved when variable is isolated in the middle all three parts must be kept balanced by doing the same operation on all parts

Example of Solving Three Part Linear Inequalities

Homework Problems Section: 1.7 Page: 156 Problems: Odd: 23 – 33 MyMathLab Assignment 31 for practice

Quadratic Inequalities Looks like a quadratic equation EXCEPT that equal sign is replaced by an inequality symbol Example:

Solving Quadratic Inequalities Put quadratic inequality in standard form (make right side zero and put trinomial in descending powers) Change quadratic inequality to a quadratic equation and solve to find “critical points” Graph “critical points” on a number line and draw a vertical line through each one to divide number line into intervals Pick a “test point” in each interval (a “nice” number that is close to zero) Evaluate the “trinomial” described in step 1 with each “test point” to determine whether the result is positive or negative and write the appropriate + or - above each test point Now graph the solution to the inequality by shading all the intervals of the number line for which the + or – satisfies the inequality written in step 1

Example of Solving Quadratic Inequality

Homework Problems Section: 1.7 Page: 157 Problems: 39 – 51 MyMathLab Assignment 32 for practice

Rational Inequality An inequality that involves a rational expression (variable in a denominator) Example:

Solving a Rational Inequality Make right side of inequality zero Perform math operations on left side to end up with a single rational expression (the rational inequality will now be in “standard form” Factor numerator and denominator of rational expression Find “critical points” by putting every factor that contains a variable equal to zero and solving Graph “critical points” on a number line and draw a vertical line through each one to divide number line into intervals Pick a “test point” in each interval (a “nice” number that is close to zero) Evaluate the left side of “standard form” described in step 1 with each “test point” to determine whether the result is positive or negative and write the appropriate + or - above each test point Now graph the solution to the problem by shading all the intervals of the number line for which the + or – satisfies inequality found in step 1

Example of Solving a Rational Inequality

Homework Problems Section: 1.7 Page: 158 Problems: Odd: 69 – 85 MyMathLab Assignment 33 for practice

Absolute Value Inequality Looks like an absolute value equation EXCEPT that an equal sign is replaced by one of the inequality symbols Examples: | 3x | – 6 > 0 | 2x – 1 | + 4 < 9 | 5x - 3 | < -7

Properties of Absolute Value | u | < 5, means that u’s distance from zero must be less than 5. Therefore, u must be located between what two numbers? between -5 and 5 How could you say this with a three part inequality? -5 < u < 5 Generalizing: | u | < n , where “n” is positive, always translates to: -n < u < n | u | > 3, means that u must be less than what, or greater than what? less than -3, or greater than 3 How could you say this with two inequalities? u < -3 or u > 3 Generalizing: | u | > n , where “n” is positive, always translates to: u < -n or u > n

Solving Absolute Value Inequalities Isolate the absolute value on the left side to write the inequality in one of the forms: | u | < n or | u | > n (where n is positive) If | u | < n, then solve: -n < u < n If | u | > n, then solve: u < -n or u > n 3. Write answer in interval notation

Example Solve: Equivalent Inequality:

Example Solve: Equivalent Inequality:

Solving Other Absolute Value Inequalities If isolating the absolute value on the left does not result in a positive number on the right side, we have to use our understanding of the definition of absolute value to come up with the solution as indicated by the following examples:

Absolute Value Inequality with No Solution How can you tell immediately that the following inequality has no solution? It says that absolute value (or distance) is negative – contrary to the definition of absolute value Absolute value inequalities of this form always have no solution:

Does this have a solution? At first glance, this is similar to the last example, because “ < 0 “ means negative, and: However, notice the symbol is: And it is possible that: We have previously learned to solve this as:

Solve this: This means that 4x – 5 can be anything except zero: Solving these two inequalities gives the solution:

Homework Problems Section: 1.8 Page: 164 Problems: Odd: 27 – 39, 45 – 61 MyMathLab Assignment 34 for practice MyMathLab Homework Quiz 8 will be due for a grade on the date of our next class meeting