EQUATIONS AND INEQUALITIES A2H CH 1 APPENDIX. Whole: 0, 1, 2, 3….. Integers: Whole numbers...

EQUATIONS AND INEQUALITIES

A2H CH 1

REAL NUMBERS AND NUMBER OPERATIONS

ALGEBRAIC EXPRESSIONS AND MODELS

SOLVING LINEAR EQUATIONS

REWRITING EQUATIONS AND FORMULAS

PROBLEM SOLVING

LINEAR INEQUALITIES

ABSOLUTE VALUE EQUATIONS AND INEQUALITIES

APPENDIX

Subsets of Real Numbers

Whole: 0, 1, 2, 3…..

Integers: Whole numbers negative and positive. … -2, -1, 0, 1, 2,

Rational Numbers: Can be written as a fraction. Either terminates or repeats.

Irrational Numbers: Can not be written as a fraction. Goes on forever without

repeating.

Real Numbers & Number Operations

A2H CH 1 Equations and Inequalities MENU

Classify each of the following. e. -5 f.

g. -4.8 h.11

INTEGER RATIONAL

RATIONAL IRRATIONAL

• Graph the real numbers , -1.8, , and -.25 and then order them from least to greatest.

-2 3-1-3 0 1 2

-.25-1.85

-1.8, -.25, , and .35

• Use the symbol < or > to show the relationship.

b. -5 and -7

a. -4 and 1

-4 < 1 or 1 > -4

-7 < -5 or -5 > -7

Example 3:

• Here are the record low temperatures for five Northeastern states.

Write the temperatures in increasing order.Connecticut -32F

Maine -48F

Maryland -40F Which states have record lows below

-40F?New Jersey -34F

Vermont -50F

Maine and Vermont

Vermont, Maine, Maryland, New Jersey, and Connecticut

PROPERTIES OF ADDTION AND MULTIPLICATIONLet a, b, and c be real numbers.

Property AdditionMultiplication CLOSURE a + b is a real number ab is a real number COMMUTATIVE a + b = b + a ab = ba ASSOCIATIVE (a + b) + c = a + (b + c) (ab)c = a(bc) IDENTITY a + 0 = a, 0 + a = a a1 = a, 1a = a INVERSE a + (-a) = 0 a (1/a) = 1 (a ≠0)

The following property involves both addition and multiplication. DISTRIBUTIVE a(b+ c) = ab + ac

235235

A INVERSE PROPERTY of ADDITION

COMMUTATIVE PROPERTY of ADDITION

INVERSE PROPERTY of MULTIPLICATION

ASSOCIATIVE PROPERTY of ADDITION

IDENTITY PROPERTY of MULTIPLICATION

State the property.

a. 8 + -8 = 0 b. 91 = 9

c. ab = ba d. 8 + (2 + 6) = (8 + 2) + 6

INVERSEaddition

ASSOCIATIVEaddition

COMMUTATIVEmultiplication

IDENTITYmultiplication

DefinitionsThe opposite (or additive inverse) of any number a is –

The reciprocal (or multiplicative inverse) of any nonzero number a is 1/a.

Subtraction is defined as adding the opposite.

Division is defined as multiplying by the reciprocal.

VOCABULARY

ADDITION

+SUBTRACTION

-MULTIPLICATION

XDIVISION

SUM DIFFERENCE PRODUCT QUOTIENT

Decreased by

Divided by Of

Less Than

More Than Minus

Increased by

For each of these operations, use the numbers in the same order they appear in

the problem

The only exception is LESS THAN. When writing a mathematical expression using LESS THAN, use the numbers in reverse

order.

a. The difference of -3 and -15 is:

b. The quotient of -18 and 2 is -9:

c. Eight less than a number is twelve.

-3 + 15 = 12

Definitions

Writing the units of each variable in a real-life problem is called unit analysis. It helps you to

determine the units for the answer.

Give the answer with the appropriate unit of measure.

a. 685 feet + 225 feet

602.25

hourshour

910 feet

2.25 dollars per pound

135 km

dollarspounds

66 60sec 60min 11sec 1min 1 5280

feet milehour feet

45 miles per hour

• You are exchanging $500 for French francs. The exchange rate is 6 francs per dollar. Assume that you use other money to pay the exchange fee.

a. How much will you receive in francs?

b. When you return you have 270 francs left. How much can

you get in dollars? Assume that you use other money to pay the exchange fee.

6500 3000

1francs

dollars francsdollar

1270 45

6dollar

francs dollarsfrancs

Exponents can be used to represent repeated multiplication.

EXPONENT

25 = 2 2 2 2 2 =

BASE FACTORS

An exponent tells you how many times the base is used as a factor.

Algebraic Expressions & Models

• Evaluate the power.

b. -34

a. (-3)4

(-3)(-3)(-3)(-3) = 81

-3333 = -81

An order of operations helps avoid confusion when evaluating

expressions.P

P Parenthesis. Compute everything inside the parenthesis.

E Exponents. Evaluate powers.

MD Multiplication/Division. Multiply and divide from left to right

AS Addition/Subtraction. Add and subtract from left to right.

• Evaluate using order of operations.

a. b. c.38 5(1 ( 3)) 4 6

2 ( 3)

2 35 6(2 ( 1))

d. e.22 4 8 3

2 10( 10) 3 ( 10)

312 19 2

125-7.5

• Evaluate using order of operations.

e. Evaluate 2x3 + 3x2 – x + 27 when x = -4

2(-4)3 + 3(-4)2 – -4 + 27 = -49

-4(-3)2 + 6(-3) – 5 = -59

d. Evaluate -4x2 + 6x – 5 when x = -3

Consider the expression 5x3 – 2x + 9.

The parts that are added or subtracted together (5x3, -2x, and 9) are called terms.

The numbers if front of the variables (5 and -2) are called coefficients of the variables.

When a term is only a number, it is called a constant (9). Terms such as 5x3 and -7x3 are like terms because they have the same variable part. Constant terms such as -6 and 4 are also like terms. You can only combine (add or subtract) like terms.

To combine like terms, add or subtract the coefficients and leave the variable and its exponent the same.

• Simplify the expression. a. -10(8 – y) – (4 – 15y) b. 4 – 3(x – 9) – (x + 1)

c. 3x + 10 – 12x – (-4) d. 3(x – 2) – 5x(x – 8)

25 - 84y -4x + 30

-9x + 145x2 + 43x - 6

Evaluate the following expression for the given values:

24 23 2 , 5, 3

xxy y x y

24 23 2 , 5, 3

xxy y x y

5 5 ( 3)

2245 18

Solving Linear Equations

4672377 xx )(

4621677 xx

171277 x17

x1260 12 12

305211 xxx5

30216 x22

3216 x16

10570120918 xx

1057012918 xx

x52234

82 x93 x

If the perimeter is 153, find the length of each side

15328293 xxx15317 x1547 x22x

Rewriting Equations & Formulas

Solving Literal Equations:

2432 yx

A Literal Equation is an equation with more than 1 variable

We frequently manipulate literal equations when working with formulas, or changing an equations “form”.

ExampleshbA 3

Find the value for y in this equation when x is 3.

Now, complete this table:

When you are asked to do something like this, it is usually easier to change

the equation.

This equation is written in standard form. Rewrite it in slope-intercept form. (solve for y)

xNow, complete this table:

xy 672

Solve the equation for the indicated variable.

whwlV ; rlrBA ;

a. You have $55 to buy digital video discs (DVDs) that cost $12 each. Write an expression for how much money you have left after buying n discs. Evaluate the expression when n = 3 and n = 4.

VERBAL MODEL

LABELS

ALGEBRAICMODEL

Price per DVD

Amount to Spend

Number of DVDs

bought

12 dollars per DVD

55 dollars

n DVDs

55 – 12n

Problem Solving

• a. Write an expression for the total monthly cost of phone service if you pay a $5 fee and 8¢ per minute. Find the cost if you talk 6 hours during the month.

VERBAL MODEL

LABELS

ALGEBRAICMODEL

Price per minute

Initial Fee

Number of Minutes

.08 dollars per minute

5 dollars n Minutes

5 + .08n

Problem Solving

Linear Inequalities

A2H CH 1 Equations and Inequalities MENUAPPENDIX

ABRIDGED ALGEBRA I

INEQUALITY NOTES:

Linear Inequalities

Tougher inequality problems:

Solve and graph:

Linear Inequalities

6 4 14 2 14x and x Solve and graph:

Linear Inequalities

4 10 2 2x or x Solve and graph:

Linear Inequalities

1 5 2 3x Solve and graph:

3-part inequalities must be “AND”

Linear Inequalities

What’s wrong with this:

Linear Inequalities

4 7x and x

Solve and graph:

No solution

Linear Inequalities

4 7x or x

Solve and graph:

ALL REAL NUMBERS

Absolute Value Equations & Inequalities

Absolute Value is how far a number is from zero on a number line.

5 This means: How far f rom zero is 5?

How far f rom zero is -5?

ABRIDGED ALGEBRA I

Absolute Value NOTES:

Solve this absolute value inequality:

3 7 1 5x

NO SOLUTION

3 7 4x

Solve this absolute value inequality:

3 7 1 5x

ALL REAL NUMBERS

3 7 4x

APPENDIX

OPENERSASSIGNMENTSEXTRA PROBLEMS

APPENDIX

REVIEW

Properties of equalityEvaluateAbsolute Value InequalityLiteral EquationsOrdering NumbersSimplifySolve (with fractions)SolveAbsolute Value Inequality

Properties of addition and multiplicationIdentify the property illustrated: 1. 4 ( 4) 0

2. ( ) ( )

3. 5 0 5

4. 5 1 5

15. 5 1

56. 3(2 1) 6 3

x y z x y z

x y is a real number

Inverse Property (addition)

Associative property (addition)Identity property

(addition)

Identity property (multiplication)

Inverse property (multiplication)

Distributive Property

REVIEW

Closure property (addition)

Evaluate the following for the values

x=4 and y=9

227. 3 ( )

3 (4 9)9 1

3 2 25

REVIEW

SOLVE FOR X

8. 9 2 3x

9 2 3 9 2 3x x 2 6x

REVIEW

Write this equation in function form (solve for y)9. 4 2 10x y

4 4x x

2 10 4y x 2 2

5 2y x

2Bygm p

A AB B

2 Ay gm p

Ay gm p

REVIEW

11. Write the following numbers in increasing order:

8137, , , 5, , 17 17

3.14159

0.42852.2360

4.7647

37 1 5 81

REVIEW

Simplify the following.

3 312. 3( 2 5) 2( )x x x 3 36 15 2 2x x x 38x 2x 15

REVIEW

SOLVE FOR X.

2 1 5 313. 5

3 2 6 4x x

2 10 5 36 3 6 4

1 10 5 33 3 6 4

4 40 10 9x x 40 6 9x 49 6x

REVIEW

SOLVE FOR X.

14. 3(2 7) 5( 3 1) 12x x

6 21x 15 5 12x

6 21 15 7x x

21 9 7x

REVIEW

Write an absolute value inequality to describe the following scenario:

15. On the Eisenhower expressway (I-290), you must

drive at least 45 mph but less than 65 mph. 55 10x

The average of the extremes The tolerance (difference between the average and extremes)

REVIEW

One of these inequalities is NO SOLUTION, the other is ALL REAL NUMBERS, can you

identify which one is which?3 8 9x

REVIEW

3 8 9x

ALL REAL NUMBERS

NO SOLUTION

EQUATIONS AND INEQUALITIES A2H CH 1 APPENDIX. Whole: 0, 1, 2, 3….. Integers: Whole numbers...

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