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

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

N

E

mph 01R

mph 7R

xx

107x7

x10

222 : hypotenuse and and legs with ngleright triaFor cbacba

x10

x7

mi 5

222 5107 xx

2510049 22 xx25149 2 x

149

252 x

149

5x

hr 149

1495x

hr 4096.0x

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?

UnknownsL RecW RecL BoxW Box .

x2x

x

2x

4

4

4

4

82 x8x

4 Box Height

LWHV 48632 xx 4481432 2 xx

19256432 2 xx1605640 2 xx

40140 2 xx 4100 xx

04 OR 010 xx

4x

10x

Impossible

.12

.10

inL

inW

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:

83 x

09923 xxx

Example One

Make one side zero:

Factor non-zero side:

Apply zero factor property and solve:

83 x

0422 2 xxx

083 x

02 x 0422 xxOR2x

2

122

12

41422 2

x

312

322i

ix

complex real-non 2 and real 1 solutions, 3 :Note

Example Two

x3 + 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 OR x2 – 9 = 0x = -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:

3

2

1

5

32

12

xxxx

Solving Rational Equations

1. Find “restricted values” for the equation by setting every denominator that contains a variable equal to zero and solving

2. Find the LCD of all the fractions and multiply both sides of equation by the LCD to eliminate fractions

3. Solve the resulting equation to find apparent solutions

4. Solutions are all apparent solutions that are not restricted

Example

3

2

1

5

32

12

xxxx

RV

01x 03 x

0322 xx

031 xx

01x03 x

OR1x 3x

SolvedAlready

SolvedAlready 3

2

1

5

31

1

xxxx

LCD 31 xx 1

LCD

3

2

1

5

31

1

xxxx

12351 xx221551 xx

1731 x

x316

3

16x RV!Not

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 = 08(x – 4)4 – 10(x – 4)2 + 3 = 0

Solving Equations that areQuadratic in Form

1. Make a substitution by letting “u” equal the repeated expression with exponent that is half of the other

2. Solve the resulting quadratic equation for “u”

3. Make a reverse substitution for “u”

4. Solve the resulting equation

Example of Solving an Equation that is Quadratic in Form

087 36 mm3Let mu

0872 uu

018 uu

08 u OR 01u8u 1u83 m 13 m

013 m083 m

0422 2 mmm

04202 2 mmorm

312 imorm

011 2 mmm

0101 2 mmorm

imorm2

3

2

11

Example of Solving an Equation that is Quadratic in Form

0341048 24 xx

03108 2 uu

24 xuLET

01234 uu

034 u 012 uOR

34 u 12 u

4

3u

2

1u

4

34 2 x

2

14 2 x

2

34 x

2

34 x

2

14 x

2

24 x

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:

21 1 x

36 12 xx ?other type what as classified be alsocan one ThisFormin Quadratic

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

21 1 x

21

1

x

RV

01x

1xLCD

1x12

1

1 LCD

x

221 x

x23

2

3x RVNot

Example of Solving Equation With Negative Integer Exponents

212 xx

22 uu

022 uu

012 uu

02 u OR 01u1u2u

1xuLet

21 x 11 x

21

x

11

x

RV0x

LCDx

12

1 LCD

x

x21

2

1x

1x

RVNot

RVNot

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:

53 xx

024 33 xx

15 xx

Solving Radical Equations

1. Isolate ONE radical on one side of the equal sign

2. Raise both sides of equation to power necessary to eliminate the isolated radical

3. Solve the resulting equation to find “apparent solutions”

4. 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:

.

etc. ,22- ,22- :however , 22 4422

etc. ,22- ,22- :and , 22 5533

Example of SolvingRadical Equation

53 xx

35 xx

22 35 xx

325102 xxx

028112 xx 074 xx

07 OR 04 xx7 OR 4 xx

4xCheck

?5344 ?514

53

7xCheck

?5377 ?547

55

solution a NOT is 4x

solution a IS 7x

Example of SolvingRadical Equation

15 xxxx 15

2215 xx

xxx 215

x24 x 2

222 x

x4

4xCheck

?1544 ?194

?132 15

solution a NOT is 4x

Solution! No hasEquation

Example of SolvingRadical Equation

024 33 xx33 24 xx

33

33 24 xx

xx 24 x4

check) toneed (No

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:

091 3

1

3

2

xx

SolvingRational Exponent Equations

• If the equation is quadratic in form, solve that way• Otherwise, solve essentially like radical equations1. Isolate ONE rational exponent expression2. Raise both sides of equation to power necessary to

change the fractional exponent into an integer exponent

3. Solve the resulting equation to find “apparent solutions”

4. 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

092 3

1

3

2

xx

3

1

3

2

92 xx

3

3

13

3

2

92

xx

xx 92 2

xxx 9442

0452 xx

014 xx

01 OR 04 xx

1 OR 4 xx

check! tohave reason to No

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 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:

3

3

u

3

3

positive.or zero is valueits However, unknown. is u"" of valuebecause ,simplified bet Can'

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

:then,052 If x 052 x

:then,352 If x 352or 352 xx

:then,352 If xx 352or 352 xxxx

:then,452 If x solution No

Example of SolvingAbsolute Value Equation

11632 x

532 x

532 x OR 532 x82 x 22 x

1x4x

Example of SolvingAbsolute Value Equation

0479 xx

479 xx

479 xx OR 479 xx

x613

x

6

13

479 xx

58 x

8

5x

Example of SolvingAbsolute Value Equation

263 x

43 x ?POSSIBLE! NOT - negative is distance says This

SOLUTION! NO hasEquation

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:

.

3315 xx inequalitypart Two

142

13 x inequalitypart Three

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

x : toequivalent is 23 x

x : toequivalent is 63 x

x : toequivalent is 82 x

5

2

4

3 Add

2-by Divide

3by Divide

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: 3315 xx

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.

2x

]2

]2,(

SolvingTwo 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

3315 xx

9315 xx

62 x

3x

!inequality of sense reverse negative, aby dividingWhen

3

]

]3,(

SolutionNotation Standard

SolutionNotation Graphical

SolutionNotation Interval

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:

.

142

13 x

142

13 x

142

13 x

Sense Same Havet Don' Symbols Inequality

1- NOT is 3-

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:

32 x

( ]2 3

]3,2(

SolvingThree 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 SolvingThree Part Linear Inequalities

142

13 x

122

13 x

246 x

22 x SolutionNotation Standard

2 2

[ ) SolutionNotation Graphical

SolutionNotation Interval)2,2[

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:

22 xx

Solving Quadratic Inequalities1. Put quadratic inequality in standard form (make right side zero and

put trinomial in descending powers)2. Change quadratic inequality to a quadratic equation and solve to

find “critical points”3. Graph “critical points” on a number line and draw a vertical line

through each one to divide number line into intervals4. Pick a “test point” in each interval (a “nice” number that is close

to zero)5. 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

6. 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 ofSolving Quadratic Inequality

22 xx

022 xx

022 xx

012 xx

01 OR 02 xx1 OR 2 xx

Numbers Critical

0 21

22 xx

:NumbersTest 2 0 3

:NumbersTest Evaluate

222 2 2002

2332 224

239

022 xxsolutions are trinomalmake that Numbers

) (

,21,

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:

31

2

x

Solving a Rational Inequality1. Make right side of inequality zero2. Perform math operations on left side to end up with a single rational

expression (the rational inequality will now be in “standard form”3. Factor numerator and denominator of rational expression4. Find “critical points” by putting every factor that contains a variable

equal to zero and solving5. Graph “critical points” on a number line and draw a vertical line through

each one to divide number line into intervals6. Pick a “test point” in each interval (a “nice” number that is close to

zero)7. 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

8. 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

11

2

x

011

2

x 0

1

1

1

1

1

2

x

x

x

01

12

x

x

01

1

x

x

01 x 01x

1x 1x

01

2 0 2

12

12

1

10

10

12

12

Numbers Critical

) [

),1[)1,(

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 5How 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 3How 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

1. 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)

2. 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:

63 x

063 x

63or 63 xx2or 2 xx

),2()2,(

Example

Solve:

Equivalent Inequality:

5125 x

9412 x

512 x

624 x

32 x

3 ,2

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 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:

275 x

)( numbernegativearepresentsnwherenu

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:

052 x

!numbernegativeathanlessbetcan'52 x

052 x

2

5

52

052

x

x

x

2

5 :isSolution x

Solve this:

• This means that 4x – 5 can be anything except zero:

• Solving these two inequalities gives the solution:

054 x

,

4

5

4

5,

054or 054 xx54or 54 xx

4

5or

4

5 xx

054or 054 xx

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