college algebra unit 1 (b) equations and inequalities.ppt

College Algebra Exam 2 Material

Quadratic ApplicationsApplication problems may give rise to all types of equations, linear, quadratic and othersHere we take a look at two that lead to quadratic equations

ExampleTwo 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 TN boy

E boy

ExampleA 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?

UnknownsL RecW RecL BoxW Box.

Homework ProblemsSection: 1.5Page:130Problems:5 9, 21 22

MyMathLab Assignment 22 for practice

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

Solving Higher Degree Polynomial EquationsSo far methods have been discussed for solving first and second degree polynomial equationsHigher 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 propertyConsider two examples:

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

Example Twox3 + x2 9x 9 = 0One side is already zero, so factor non-zero sidex3 + x2 9x 9 = 0x2(x + 1) 9(x + 1) = 0(x + 1)(x2 9) = 0Apply zero factor property and solve:x + 1 = 0 ORx2 9 = 0x = -1x2 = 9x = 3

Homework ProblemsSection: 1.4Page:124Problems:59 62


Rational EquationsTechnical Definition: An equation that contains a rational expressionPractical Definition: An equation that has a variable in a denominatorExample:

Solving Rational EquationsFind 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 fractionsSolve the resulting equation to find apparent solutionsSolutions are all apparent solutions that are not restricted

Example

Homework ProblemsSection:1.6Page:144Problems:Odd: 1 25


Quadratic in Form EquationAn 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 timeExamples: m6 7m3 8 = 08(x 4)4 10(x 4)2 + 3 = 0

Solving Equations that areQuadratic in FormMake a substitution by letting u equal the repeated expression with exponent that is half of the otherSolve the resulting quadratic equation for uMake a reverse substitution for uSolve the resulting equation

Example of Solving an Equation that is Quadratic in Form

Homework ProblemsSection:1.6Page:145Problems:All: 61 64, 73 74


Negative Integer Exponent EquationAn equation is a negative integer exponent equation if it has a variable expression with a negative integer exponentExamples:

Solving Negative Integer Exponent EquationsIf 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

Homework ProblemsSection:1.6Page:145Problems:75, 76

MyMathLab Assignment 26 for practiceMyMathLab Homework Quiz 5/6 will be due for a grade on the date of our next class meeting

Radical EquationsAn equation is called a radical equation if it contains a variable in a radicandExamples:

Solving Radical EquationsIsolate ONE radical on one side of the equal signRaise both sides of equation to power necessary to eliminate the isolated radicalSolve the resulting equation to find apparent solutionsApparent 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 equationsIf both sides of equation are raised to an odd power, then resulting equations are equivalentIf 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 SolvingRadical Equation

Homework ProblemsSection:1.6Page:144Problems:Odd: 27 51, 55 57


Rational Exponent EquationsAn equation in which a variable expression is raised to a fractional powerExample:

SolvingRational Exponent EquationsIf the equation is quadratic in form, solve that wayOtherwise, solve essentially like radical equationsIsolate ONE rational exponent expressionRaise both sides of equation to power necessary to change the fractional exponent into an integer exponentSolve the resulting equation to find apparent solutionsApparent 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 ProblemsSection:1.6Page:145Problems:All: 53 54, 59 60, 65 72


Definition of Absolute ValueAbsolute value means distance away from zero on a number lineDistance is always positive or zeroAbsolute value is indicated by placing vertical parallel bars on either side of a number or expressionExamples: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 EquationAn equation that has a variable contained within absolute value symbolsExamples:| 2x 3 | + 6 = 11| x 8 | | 7x + 4 | = 0| 3x | + 4 = 0

Solving Absolute Value EquationsIsolate one absolute value that contains an algebraic expression, | u |If the other side is a negative number there is no solution (distance cant 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 SolvingAbsolute Value Equation

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

InequalitiesAn equation is a comparison that says two algebraic expressions are equalAn 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:

.

InequalitiesThere are lots of different types of inequalities, and each is solved in a special wayInequalities are called equivalent if they have exactly the same solutionsEquivalent inequalities are obtained by using properties of inequalities

Properties of InequalitiesAdding 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 InequalitiesWhereas solutions to equations are usually sets of individual numbers, solutions to inequalities are typically intervals of numbersExample:Solution to x = 3 is {3}Solution to x < 3 is every real number that is less than threeSolutions to inequalities may be expressed in:Standard NotationGraphical NotationInterval Notation

Two Part Linear InequalitiesA 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 InequalitiesStandard 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.

SolvingTwo Part Linear InequalitiesSolve exactly like linear equations EXCEPT:Always isolate variable on left side of inequalityCorrectly 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

Three Part Linear InequalitiesConsist of three algebraic expressions compared with two inequality symbolsBoth inequality symbols MUST have the same sense (point the same direction) AND must make a true statement when the middle expression is ignoredGood Example:

Not Legitimate:

.

Expressing Solutions to Three Part InequalitiesStandard 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:

SolvingThree Part Linear InequalitiesSolved exactly like two part linear inequalities except that:solution is achieved when variable is isolated in the middleall three parts must be kept balanced by doing the same operation on all parts

Example of SolvingThree Part Linear Inequalities

Quadratic InequalitiesLooks like a quadratic equation EXCEPT that equal sign is replaced by an inequality symbolExample:

Solving Quadratic InequalitiesPut 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 pointsGraph critical points on a number line and draw a vertical line through each one to divide number line into intervalsPick 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 pointNow 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 ofSolving Quadratic Inequality

Homework ProblemsSection:1.7Page:157Problems:39 51


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

Solving a Rational InequalityMake right side of inequality zeroPerform math operations on left side to end up with a single rational expression (the rational inequality will now be in standard formFactor numerator and denominator of rational expressionFind critical points by putting every factor that contains a variable equal to zero and solvingGraph critical points on a number line and draw a vertical line through each one to divide number line into intervalsPick 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 pointNow 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

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

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

Solving Absolute Value InequalitiesIsolate 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

ExampleSolve:

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 Inequalitywith No SolutionHow 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 valueAbsolute 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 ProblemsSection:1.8Page:164Problems:Odd: 27 39, 45 61


MyMathLab Homework Quiz 8 will be due for a grade on the date of our next class meeting

college algebra unit 1 (b) equations and inequalities.ppt

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