Review Topics(Ch R & 1 in College Algebra Book)• Exponents & Radical Expressions (P. 21-25 and P. 72-77)

• Complex Numbers (P. 109 – 114)

• Factoring (p. 49 – 55)

• Quadratic Equations (P. 97 – 105)

• Rational Expressions (P. 61 – 69)

• Rational Equations & Clearing Fractions (P. 88 – 91)

• Radical Equations (P. 118 – 123)

• 1.5: Solving Inequalities

• 1.6: Equations and Inequalities involving absolute value

Review of Exponents82 =8 • 8 = 64 24 = 2 • 2 • 2 • 2 = 16

x2 = x • x x4 = x • x • x • x Base = x Base = xExponent = 2 Exponent = 4

Exponents of 1 Zero ExponentsAnything to the 1 power is itself Anything to the zero power = 1

51 = 5 x1 = x (xy)1 = xy 50 = 1 x0 = 1 (xy)0 = 1

Negative Exponents

5-2 = 1/(52) = 1/25 x-2 = 1/(x2) xy-3 = x/(y3) (xy)-3 = 1/(xy)3 = 1/(x3y3)

a-n = 1/an 1/a-n = an a-n/a-m = am/an

Powers with Base 10100 = 1101 = 10102 = 100103 = 1000104 = 10000

The exponent is the same as the The exponent is the same as the numbernumber of 0’s after the 1. of digits after the decimal where 1 is placed

100 = 110-1 = 1/101 = 1/10 = .110-2 = 1/102 = 1/100 = .0110-3 = 1/103 = 1/1000 = .00110-4 = 1/104 = 1/10000 = .0001

Scientific Notation uses the concept of powers with base 10.

Scientific Notation is of the form: __. ______ x 10(** Note: Only 1 digit to the left of the decimal)

You can change standard numbers to scientific notationYou can change scientific notation numbers to standard numbers

Scientific NotationScientific Notation uses the concept of powers with base 10.

Scientific Notation is of the form: __. ______ x 10(** Note: Only 1 digit to the left of the decimal)

-25 321

Changing a number from scientific notation to standard formStep 1: Write the number down without the 10n part.Step 2: Find the decimal pointStep 3: Move the decimal point n places in the ‘number-line’ direction of the sign of the exponent.Step 4: Fillin any ‘empty moving spaces’ with 0.

Changing a number from standard form to scientific notationStep1: Locate the decimal point.Step 2: Move the decimal point so there is 1 digit to the left of the decimal.Step 3: Write new number adding a x 10n where n is the # of digits moved left adding a x10-n where n is the #digits moved right

5.321

.05321

.0 5 3 2 1= 5.321 x 10-2

Raising Quotients to Powers

a n

b = an

bna -n b

= a-n

b-n= bn

an= b n

a

Examples: 3 2 32 94 42 16= =

2x 3 (2x)3 8x3

y y3 y3= =

2x -3 (2x)-3 1 y3 y3

y y-3 y-3(2x)3 (2x)3 8x3= = = =

Product Ruleam • an = a(m+n)

x3 • x5 = xxx • xxxxx = x8

x-3 • x5 = xxxxx = x2 = x2

xxx 1

x4 y3 x-3 y6 = xxxx•yyy•yyyyyy = xy9 xxx

3x2 y4 x-5 • 7x = 3xxyyyy • 7x = 21x-2 y4 = 21y4

xxxxx x2

Quotient Rule

am = a(m-n)

an

43 = 4 • 4 • 4 = 41 = 4 43 = 64 = 8 = 442 4 • 4 42 16 2

x5 = xxxxx = x3 x5 = x(5-2) = x3

x2 xx x2

15x2y3 = 15 xx yyy = 3y2 15x2y3 = 3 • x -2 • y2 = 3y2 5x4y 5 xxxx y x2 5x4y x2

3a-2 b5 = 3 bbbbb bbb = b8 3a-2 b5 = a(-2-4)b(5-(-3)) = a-6 b8 = b8

9a4b-3 9aaaa aa 3a6 9a4b-3 3 3 3a6

Powers to Powers

(am)n = amn

(a2)3 a2 • a2 • a2 = aa aa aa = a6

(24)-2 = 1 = 1 = 1 = 1/256 (24)2 24 • 24 16 • 16 28 256

(x3)-2 = x –6 = x 10 = x4

(x -5)2 x –10 x 6

(24)-2 = 2-8 = 1 = 1

Products to Powers

(ab)n = anbn

(6y)2 = 62y2 = 36y2

(2a2b-3)2 = 22a4b-6 = 4a4 = a4(ab3)3 4a3b9 4a3b9b6 b15

What about this problem?

5.2 x 1014 = 5.2/3.8 x 109 1.37 x 109

3.8 x 105

Do you know how to do exponents on the calculator?

Square Roots & Cube Roots

A number b is a square root of a number a if b2 = a

25 = 5 since 52 = 25

Notice that 25 breaks down into 5 • 5So, 25 = 5 • 5

See a ‘group of 2’ -> bring it outside theradical (square root sign).

Example: 200 = 2 • 100 = 2 • 10 • 10 = 10 2

A number b is a cube root of a number a if b3 = a

8 = 2 since 23 = 8

Notice that 8 breaks down into 2 • 2 • 2 So, 8 = 2 • 2 • 2

See a ‘group of 3’ –> bring it outsidethe radical (the cube root sign)

Example: 200 = 2 • 100 = 2 • 10 • 10 = 2 • 5 • 2 • 5 • 2

= 2 • 2 • 2 • 5 • 5 = 2 25

3

3

3 3

3

3

3

3

Note: -25 is not a real number since nonumber multiplied by itself will be negative

Note: -8 IS a real number (-2) since-2 • -2 • -2 = -8

3

Nth Root ‘Sign’ Examples

16

-16

= 4 or -4

not a real number

-164

not a real number

Even radicals of negative numbersAre not real numbers.

-325

= -2 Odd radicals of negative numbersHave 1 negative root.

325

= 2 Odd radicals of positive numbersHave 1 positive root.

Even radicals of positive numbersHave 2 roots. The principal rootIs positive.

Exponent Rules( )x x

x x x

x

xx

m n mn

m n m n

m

nm n

x

xx

x x

mm

m m

0

1

1

1

/

(XY)m = xmym

XY

m

=Xm

Ym

Examples to Work through

3 34

4

3

8

12

81

27

yx

Product Rule and Quotient Rule Example

4/1

4/34/5

8

88

Some Rules for Simplifying Radical Expressions

nmn m

nn

nnn

aa

aa

abba

/

/1

Example Set 1

300

162

75

55

33

x

y

y

x

Example Set 2

4 4

3

512

54

16

x

Example Set 3

55

56

5

6

33

27

8

9

4

84

1255

r

t

r

t

tt

Operations on Radical Expressions

•Addition and Subtraction (Combining LIKE Terms)

•Multiplication and Division

• Rationalizing the Denominator

Radical Operations with Numbers

333 210545162

2423

Radical Operations with Variables

zzz

yxxy

xx

48312332

3

2

27

8

4 54 5

3

3

Multiplying Radicals (FOIL works with Radicals Too!)

)8)(9(

)32)(32(

xx

yxyx

Rationalizing the Denominator

• Remove all radicals from the denominator

3

2

1

y

xy

Rationalizing Continued…

• Multiply by the conjugate

23

3

23

1

Complex Numbers

REAL NUMBERS Imaginary Numbers

IrrationalNumbers

, 8, -13

Rational Numbers(1/2 –7/11, 7/9, .33

Integers(-2, -1, 0, 1, 2, 3...)

Whole Numbers(0,1,2,3,4...)

Natural Numbers(1,2,3,4...)

Complex Numbers(a + bi)

Real Numbersa + bi with b = 0

Imaginary Numbersa + bi with b 0

i = -1 where

i2= -1

IrrationalNumbers

Rational Numbers

Integers

Whole Numbers

Natural Numbers

Simplifying Complex NumbersA complex number is simplified if it is in standard form:

a + bi

Addition & Subtraction)Ex1: (5 – 11i) + (7 + 4i) = 12 – 7i

Ex2: (-5 + 7i) – (-11 – 6i) = -5 + 7i +11 + 6i = 6 + 13i

Multiplication)Ex3: 4i(3 – 5i) = 12i –20i2 = 12i –20(-1) = 12i +20 = 20 + 12i

Ex4: (7 – 3i) (-2 – 5i) [Use FOIL] -14 –35i +6i +15i2

-14 –29i +15(-1) -14 –29i –15 -29 –29i

Complex ConjugatesThe complex conjugate of (a + bi) is (a – bi)The complex conjugate of (a – bi) is (a + bi)

(a + bi) (a – bi) = a2 + b2

Division7 + 4i2 – 5i

2 + 5i 14 + 35i + 8i + 20i2 14 + 43i +20(-1)2 + 5i 4 + 10i –10i – 25i2 4 –25(-1)

14 + 43i –20 -6 + 43i -6 434 + 25 29 29 29

= =

= + i=

Square Root of a Negative Number

25 4 = 100 = 10

-25 -4 = (-1)(25) (-1)(4)

= (i2)(25) (i2)(4) = i 25 i 4 = (5i) (2i) = 10i2 = 10(-1) = -10

Optional Step

Practice – Square Root of Negatives

i 1

12

16

4

Practice – Simplify Imaginary Numbers

i2 =

i3 =

i4 =

i5 =

i6 =

-1

-i

1

i

-1

i0 = 1i1 = i

Another way to calculate in

Divide n by 4. If the remainder is rthen in = ir

Example:i11 = __________

11/4 = 2 remainder 3

So, i11 = i3 = -i

Practice – Simplify More Imaginary Numbers

203

100

26

15

i

i

i

i

Practice – Addition/Subtraction

)7()93(

)7()93(

ii

ii 10 +8i

-4 +10i

Practice – Complex Conjugates

• Find complex conjugate.

i

i

43

25

3i =>

-4i =>

Practice Division w/Complex Conjugates

i

i

4

47

4__2i

=

Adding & Subtracting Polynomials

Combine Like Terms

(2x2 –3x +7) + (3x2 + 4x – 2) = 5x2 + x + 5

(5x2 –6x + 1) – (-5x2 + 3x – 5) = (5x2 –6x + 1) + (5x2 - 3x + 5) = 10x2 – 9x + 6

Types of Polynomialsf(x) = 3 Degree 0 Constant Functionf(x) = 5x –3 Degree 1 Linear f(x) = x2 –2x –1 Degree 2 Quadraticf(x) = 3x3 + 2x2 – 6 Degree 3 Cubic

Multiplication of Polynomials

Step 1: Using the distributive property, multiply every term in the 1st polynomial by every term in the 2nd polynomial

Step 2: Combine Like TermsStep 3: Place in Decreasing Order of Exponent

4x2 (2x3 + 10x2 – 2x – 5) = 8x5 + 40x4 –8x3 –20x2

(x + 5) (2x3 + 10x2 – 2x – 5) = 2x4 + 10x3 – 2x2 – 5x + 10x3 + 50x2 – 10x – 25

= 2x4 + 20x3 + 48x2 –15x -25

Binomial Multiplication with FOIL

(2x + 3) (x - 7)

F. O. I. L.(First) (Outside) (Inside) (Last)

(2x)(x) (2x)(-7) (3)(x) (3)(-7)

2x2 -14x 3x -21

2x2 -14x + 3x -21

2x2 - 11x -21

Division by a Monomial3x2 + x 5x3 – 15x2

x 15x

4x2 + 8x – 12 5x2y + 10xy2

4x2 5xy

15A2 – 8A2 + 12 12A5 – 8A2 + 12 4A 4A

Review: Factoring Polynomials

To factor a polynomial, follow a similar process.

Factor: 3x4 – 9x3 +12x2

3x2 (x2 – 3x + 4)

To factor a number such as 10, find out

‘what times what’ = 10

10 = 5(2)

Another Example:Factor 2x(x + 1) + 3 (x + 1)

(x + 1)(2x + 3)

Solving Polynomial Equations By Factoring

Solve the Equation: 2x2 + x = 0

Step 1: Factor x (2x + 1) = 0

Step 2: Zero Product x = 0 or 2x + 1 = 0

Step 3: Solve for X x = 0 or x = - ½

Zero Product Property : If AB = 0 then A = 0 or B = 0

Question: Why are there 2 values for x???

Factoring Trinomials

To factor a trinomial means to find 2 binomials whose productgives you the trinomial back again.

Consider the expression: x2 – 7x + 10

(x – 5) (x – 2)The factored form is:

Using FOIL, you can multiply the 2 binomials andsee that the product gives you the original trinomial expression.

How to find the factors of a trinomial:

Step 1: Write down 2 parentheses pairs.Step 2: Do the FIRSTSStep3 : Do the SIGNSStep4: Generate factor pairs for LASTSStep5: Use trial and error and check with FOIL

Practice

Factor:

1. y2 + 7y –30 4. –15a2 –70a + 120

2. 10x2 +3x –18 5. 3m4 + 6m3 –27m2

3. 8k2 + 34k +35 6. x2 + 10x + 25

Special Types of FactoringSquare Minus a Square

A2 – B2 = (A + B) (A – B)

Cube minus Cube and Cube plus a Cube

(A3 – B3) = (A – B) (A2 + AB + B2)

(A3 + B3) = (A + B) (A2 - AB + B2)

Perfect Squares

A2 + 2AB + B2 = (A + B)2

A2 – 2AB + B2 = (A – B)2

Quadratic Equations

General Form of Quadratic Equation

ax2 + bx + c = 0 a, b, c are real numbers & a 0

A quadratic Equation: x2 – 7x + 10 = 0 a = _____ b = _____ c = ______

Methods & Tools for Solving Quadratic Equations1. Factor 2. Apply zero product principle (If AB = 0 then A = 0 or B = 0)3. Square root method4. Completing the Square5. Quadratic Formula

Example1: Example 2:x2 – 7x + 10 = 0 4x2 – 2x = 0(x – 5) (x – 2) = 0 2x (2x –1) = 0x – 5 = 0 or x – 2 = 0 2x=0 or 2x-1=0 + 5 + 5 + 2 + 2 2 2 +1 +1

2x=1x = 5 or x = 2 x = 0 or x=1/2

1 -7 10

Square Root Method

If u2 = d then u = d or u = - d. If u2 = d then u = + d

Solving a Quadratic Equation with the Square Root MethodExample 1: Example 2:4x2 = 20 (x – 2)2 = 64 4

x – 2 = +6 x2 = 5 + 2 + 2

x = + 5 x = 2 + 6

So, x = 5 or - 5 So, x = 2 + 6 or 2 - 6

Completing the Square

If x2 + bx is a binomial then by adding b 2 which is the square of half 2

the coefficient of x, a perfect square trinomial results:

x2 + bx + b 2 = x + b 2

2 2

Solving a quadratic equation with ‘completing the square’ method.

Example: Step1: Isolate the Binomialx2 - 6x + 2 = 0 -2 -2 Step 2: Find ½ the coefficient of x (-3 )x2 - 6x = -2 and square it (9) & add to both sides.x2 - 6x + 9 = -2 + 9(x – 3)2 = 7x – 3 = + 7

x = (3 + 7 ) or (3 - 7 )

Note: If the coefficient of x2 is not 1 you must divide by the coefficient of x2 beforecompleting the square. ex: 3x2 – 2x –4 = 0(Must divide by 3 before taking ½ coefficient of x)

Step 3: Apply square root method

(Example 1)

(Completing the Square – Example 2)

2x2 +4x – 1 = 0

2x2 +4x – 1 = 0 (x + 1) (x + 1) = 3/2 2 2 2 2 (x + 1)2 = 3/2x2 +2x – 1/2 = 0 (x2 +2x ) = ½ √(x + 1)2 = √3/2(x2 +2x + 1 ) = 1/2 + 1 x + 1 = +/- √6/2

x = √6/2 – 1 or - √6/2 - 1

Step 1: Check the coefficient of the x2 term. If 1 goto step 2 If not 1, divide both sides by the coefficient of the x2 term.

Step 2: Calculate the value of : (b/2)2 [In this example: (2/2)2 = (1)2 = 1]

Step 3: Isolate the binomial by grouping the x2 and x term together, then add (b/2)2 to both sides of he equation.

Step 4: Factor & apply square root method

Quadratic FormulaGeneral Form of Quadratic Equation: ax2 + bx + c = 0

Quadratic Formula: x = -b + b2 – 4ac discriminant: b2 – 4ac 2a if 0, one real solution if >0, two unequal real solutions if <0, imaginary solutionsSolving a quadratic equation with the ‘Quadratic Formula’

2x2 – 6x + 1= 0 a = ______ b = ______ c = _______

x = - (-6) + (-6)2 – 4(2)(1) 2(2)

= 6 + 36 –8 4

= 6 + 28 = 6 + 27 = 2 (3 + 7 ) = (3 + 7 ) 4 4 4 2

2 -6 1

Solving Higher Degree Equations

x3 = 4x

x3 - 4x = 0x (x2 – 4) = 0x (x – 2)(x + 2) = 0

x = 0 x – 2 = 0 x + 2 = 0

x = 2 x = -2

2x3 + 2x2 - 12x = 0

2x (x2 + x – 6) = 0

2x (x + 3) (x – 2) = 0

2x = 0 or x + 3 = 0 or x – 2 = 0

x = 0 or x = -3 or x = 2

Solving By Grouping

x3 – 5x2 – x + 5 = 0

(x3 – 5x2) + (-x + 5) = 0

x2 (x – 5) – 1 (x – 5) = 0

(x – 5)(x2 – 1) = 0

(x – 5)(x – 1) (x + 1) = 0

x – 5 = 0 or x - 1 = 0 or x + 1 = 0

x = 5 or x = 1 or x = -1

Rational Expressions

Rational Expression – an expression in which a polynomial is divided by another nonzero polynomial.

Examples of rational expressions

4 x 2x 2x – 5 x – 5

Domain = {x | x 0} Domain = {x | x 5/2} Domain = {x | x 5}

Multiplication and Division of Rational Expressions

A • C = A 9x = 3B • C B 3x2 x

5y – 10 = 5 (y – 2) = 5 = 110y - 20 10 (y – 2) 10 2

2z2 – 3z – 9 = (2z + 3) (z – 3) = 2z + 3z2 + 2z – 15 (z + 5) (z – 3) z + 5

A2 – B2 = (A + B)(A – B) = (A – B)A + B (A + B)

Negation/Multiplying by –1

-y – 24y + 8

- = y + 2 4y + 8 OR -y - 2

-4y - 8

Examples

x3 – x x + 1x – 1 x

•

(x3 – x) (x + 1) x(x – 1)=

x (x2 – 1)(x + 1) x(x – 1)

=

= x (x + 1) (x – 1)(x + 1) x(x – 1)

= (x + 1)(x + 1) = (x + 1)2

x2 – 25 x2 –10x + 25x2 + 5x + 4 2x2 + 8x

=x2 – 25 2x2 + 8xx2 + 5x + 4 x2 –10x + 25

•

=(x + 5) (x – 5) • 2x(x + 4)(x + 4)(x + 1) • (x – 5) (x – 5)

=2x (x + 5)(x + 1)(x – 5)

Check Your Understanding

Simplify:

x2 –6x –7 x2 -1

Simplify:

1 3x - 2 x2 + x - 6

(x + 1) (x –7)(x + 1) (x – 1)

(x – 7)(x – 1)

1 x2 + x - 6x – 2 3•

1 (x + 3) (x – 2)x – 2 3

•

(x + 3) 3

Addition of Rational ExpressionsAdding rational expressions is like adding fractions

With LIKE denominators:

1 + 2 = 3 8 8 8

x + 3x - 1 = 4x - 1 x + 2 x + 2 x + 2

x + 2 (2 + x) (2 + x)3x2 + 4x - 4 3x2 + 4x -4 (3x2 + 4x – 4) (3x -2)(x + 2)

= =

= 1 (3x – 2)

Adding with UN-Like Denominators

3 + 14 8

(3) (2) + 18 8

6 + 18 8

7 8

1 + 2x2 – 9 x + 3

1 + 2(x + 3)(x – 3) (x + 3)

1 + 2 (x – 3)(x + 3)(x – 3) (x + 3)(x – 3)

1 + 2(x – 3) 1 + 2x – 6 2x - 5(x + 3) (x – 3) (x + 3) (x – 3) (x + 3) (x – 3)

= =

Subtraction of Rational Expressions

2x - x + 1x2 – 1 x2 - 1

To subtract rational expressions:Step 1: Get a Common DenominatorStep 2: Combine Fractions DISTRIBUTING the ‘negative sign’

BE CAREFUL!!

=2x – (x + 1)x2 -1

= x – 1(x + 1)(x –1)

= 1(x + 1)

= 2x – x - 1x2 -1

Check Your Understanding

Simplify:

b b-12b - 4 b-2-

b b-12(b – 2) b-2

-

b -b+12(b – 2) b-2

+

b2(b – 2)

2(-b+1)2(b – 2)

+

b –2b+22(b – 2)

-b + 22(b – 2)= =

-1(b – 2)2(b – 2)

= -12

Complex Fractions

A complex fraction is a rational expression that contains fractions in its numerator, denominator, or both.

Examples:

15

47

xx2 – 16

1x - 4

1x

+ 2x2

3x

- 1x2

7/20 xx + 4

x + 23x - 1

Rational Equations

3x = 3 x + 1 = 3 6 = x2x – 1 x – 2 x - 2 x + 1

(2x – 1)

3x = 3(2x – 1)3x = 6x – 3-3x = -3

x = 1

(x - 2)

x + 1 = 3

x = 2

(x + 1)

6 = x (x + 1)

6 = x2 + x

x2 + x – 6 = 0

(x + 3 ) (x - 2 ) = 0

x = -3 or x = 2Careful! – What doYou notice about the answer?

Rational Equations Cont…To solve a rational equation:

Step 1: Factor all polynomialsStep 2: Find the common denominatorStep 3: Multiply all terms by the common denominatorStep 4: Solve

x + 1 - x – 1 = 1 2x 4x 3

(12x)

= 6 (x + 1) -3(x – 1) = 4x6x + 6 –3x + 3 = 4x

3x + 9 = 4x -3x -3x 9 = x

Other Rational Equation Examples

3 + 5 = 12x – 2 x + 2 x2 - 4

3 + 5 = 12x – 2 x + 2 (x + 2) (x – 2)

(x + 2)(x – 2)

3(x + 2) + 5(x – 2) = 12

3x + 6 + 5x – 10 = 12

8x – 4 = 12 + 4 + 4

8x = 16

x = 2

1 + 1 = 3x x2 4

(4x2)

4x + 4 = 3x2

3x2 - 4x - 4 = 0

(3x + 2) (x – 2) = 0

3x + 2 = 0 or x – 2 = 0

3x = -2 or x = 2

x = -2/3 or x = 2

Check Your Understanding

Simplify:x 1x2 – 1 x2 – 1

1 3x – 2 x

1 1 2x(x – 1) x2 – 1 x(x + 1)

Solve6 1x 2

3 22x – 1 x + 1

2 3 xx – 1 x + 2 x2 + x - 2

+

-

+ -

- = 1

=

+ =

1x - 1

2(x – 3)x(x – 2)

3x(x – 1)(x + 1)

4

5

-1/4

1 = 1 + 1F p q

Solve for p:Try this one:

Solving Radical Equations

25)63( 2 xX2 = 64

10003 x

1000)4( 3 x

#1

#2

#3

#4

Radical Equations Continued…

Example 1:

x + 26 – 11x = 4

26 – 11x = 4 - x

(26 – 11x)2 = (4 – x)2

26 – 11x = (4-x) (4-x)

26 - 11x = 16 –4x –4x +x2

26 –11x = 16 –8x + x2

-26 +11x -26 +11x0 = x2 + 3x -100 = (x - 2) (x + 5) x – 2 = 0 or x + 5 = 0 x = 2 x = -5

Example 2:

3x + 1 – x + 4 = 1

3x + 1 = x + 4 + 1

(3x + 1)2 = (x + 4 + 1)2

3x + 1 = (x + 4 + 1) (x + 4 + 1)

3x + 1 = x + 4 + x + 4 + x + 4 + 13x + 1 = x + 4 + 2x + 4 + 13x + 1 = x + 5 + 2x + 4 -x -5 -x -5 2x - 4 = 2x + 4 (2x - 4)2 = (2x + 4)2

4x2 –16x +16 = 4(x+4) 4x2 –20x = 0 4x(x –5) = 0, so…4x = 0 or x – 5 = 0 x = 0 or x = 5

4x+16

1.5 Inequality Set & Interval Notation

Set Builder Notation{1,5,6} { } {6}

{x | x > -4} {x | x < 2} {x | -2 < x < 7}x such that x such that x is less x such that x is greaterx is greater than –4 than or equal to 2 than –2 and less than or equal to 7

Interval (-4, ) (-, 2] (-2, 7]Notation

Graph-4 2 -2 0 7

Question: How would you write the set of all real numbers? (-, ) or R

Inequality Example

Statement Reason7x + 15 > 13x + 51 [Given]

-6x + 15 > 51 [-13x]

-6x > 36 [-15]

x < 6 [Divide by –6, so must ‘flip’ the inequality sign

Set Notation: {x | x < 6}

Interval Notation: (-, 6]

Graph:

6

Compound Inequality

-3 < 2x + 1 < 3 Set Notation: {x | -2 < x < 1}

-1 -1 -1-4 < 2x < 2 Interval Notation: (-2, 1]

2 2 2Graph:

-2 < x < 1 -2 0 1

Set Operations and Compound Inequalities

Union () – “OR”A B = {x | x A or x B}

-4x + 1 9 or 5X+ 3 12 X -2 or X -3

Intersection () – “AND”A B = {x | x A and x B}

X+ 1 9 and X – 2 3

X 8 and X 5

Set Notation: {x | X 8 and X 5}

Interval Notation: (- , 8] [5, )

0 5 8

[ ]

Set Notation: {x | X -2 or X -3}

Interval Notation: (- , -2] (- , -3]

-2

1.6 Absolute Value Inequality

| 2x + 3| > 5

2x + 3 > 5 or -(2x + 3) > 5

2x > 2 -2x - 3 > 5

x > 1 -2x > 8 -2 -2

x < -4

Set Notation: {x | x < -4 or x > 1}

Interval Notation: (-, -4] or [1, )

Graph: -4 0 1

Absolute Value Equations

| 2x – 3| = 11

2x – 3 = 11 or -(2x – 3) = 11

2x = 14 -2x + 3 = 11

x = 7 -2x = 8

x = -4