EQUATIONS AND INEQUALITIES A2H CH 1 APPENDIX

Whole: 0, 1, 2, 3….. Integers: Whole numbers negative and positive. … -2, -1, 0, 1, 2, 3… 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.-5f. g. -4.8h. INTEGER RATIONAL IRRATIONAL Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

Graph the real numbers, -1.8,, and -.25 and then order them from least to greatest , -.25,, and. Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

Use the symbol to show the relationship. b.-5 and -7 a.-4 and Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

Example 3: Here are the record low temperatures for five Northeastern states. Write the temperatures in increasing order. Connecticut-32  F Maine-48  F Maryland-40  F Which states have record lows below -40  F? New Jersey-34  F Vermont-50  F Maine and Vermont Vermont, Maine, Maryland, New Jersey, and Connecticut Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

Let a, b, and c be real numbers. PropertyAdditionMultiplication CLOSUREa + b is a real numberab is a real number COMMUTATIVEa + b = b + aab = ba ASSOCIATIVE(a + b) + c = a + (b + c)(ab)c = a(bc) IDENTITYa + 0 = a, 0 + a = aa  1 = a, 1  a = a INVERSEa + (-a) = 0a  (1/a) = 1 (a ≠0) The following property involves both addition and multiplication. DISTRIBUTIVEa(b+ c) = ab + ac Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

INVERSE PROPERTY of ADDITION COMMUTATIVE PROPERTY of ADDITION INVERSE PROPERTY of MULTIPLICATION ASSOCIATIVE PROPERTY of ADDITION IDENTITY PROPERTY of MULTIPLICATION Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

 State the property. a = 0b. 91 = 9 c. ab = bad. 8 + (2 + 6) = (8 + 2) + 6 INVERSE addition ASSOCIATIVE addition COMMUTATIVE multiplication IDENTITY multiplication Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

The opposite (or additive inverse) of any number a is –a. 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. Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

VOCABULARY ADDITION + SUBTRACTION - MULTIPLICATION X DIVISION ÷ SUMDIFFERENCEPRODUCTQUOTIENT Decreased by Divided by Of Less Than More ThanMinus Times 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. Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

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 = 12 = -9 Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

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. Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

 Give the answer with the appropriate unit of measure. a.685 feet feet b. c. d. 910 feet 2.25 dollars per pound 135 km 45 miles per hour Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

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. Real Numbers & Number Operations A2H CH 1 Equations and Inequalities MENU

Exponents can be used to represent repeated multiplication. EXPONENT 2 5 = 2  2  2  2  2 = BASE FACTORS An exponent tells you how many times the base is used as a factor. Algebraic Expressions & Models A2H CH 1 Equations and Inequalities MENU

Evaluate the power. b a. (-3) 4 (-3)(-3)(-3)(-3) = 81 -3333 = -81 Algebraic Expressions & Models A2H CH 1 Equations and Inequalities MENU

An order of operations helps avoid confusion when evaluating expressions. P E MD AS 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. Algebraic Expressions & Models A2H CH 1 Equations and Inequalities MENU

Evaluate using order of operations. a. b.c. d. e. Algebraic Expressions & Models A2H CH 1 Equations and Inequalities MENU

Evaluate using order of operations. e. Evaluate 2x 3 + 3x 2 – x + 27 when x = -4 2(-4) 3 + 3(-4) 2 – = (-3) 2 + 6(-3) – 5 = -59 d. Evaluate -4x 2 + 6x – 5 when x = -3 Algebraic Expressions & Models A2H CH 1 Equations and Inequalities MENU

Algebraic Expressions & Models A2H CH 1 Equations and Inequalities MENU Consider the expression 5x 3 – 2x + 9. The parts that are added or subtracted together (5x 3, -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 5x 3 and -7x 3 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) Algebraic Expressions & Models A2H CH 1 Equations and Inequalities MENU y-4x x x x - 6

Evaluate the following expression for the given values: Algebraic Expressions & Models A2H CH 1 Equations and Inequalities MENU

Solving Linear Equations A2H CH 1 Equations and Inequalities MENU

Solving Linear Equations A2H CH 1 Equations and Inequalities MENU

Solving Linear Equations A2H CH 1 Equations and Inequalities MENU

Solving Linear Equations A2H CH 1 Equations and Inequalities MENU If the perimeter is 153, find the length of each side

Rewriting Equations & Formulas A2H CH 1 Equations and Inequalities MENU Solving Literal Equations: 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”. Examples

Rewriting Equations & Formulas A2H CH 1 Equations and Inequalities MENU Find the value for y in this equation when x is 3. Now, complete this table: XY When you are asked to do something like this, it is usually easier to change the equation.

Rewriting Equations & Formulas A2H CH 1 Equations and Inequalities MENU This equation is written in standard form. Rewrite it in slope-intercept form. (solve for y) Now, complete this table: XY

Rewriting Equations & Formulas A2H CH 1 Equations and Inequalities MENU Solve the equation for the indicated variable.

 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 ALGEBRAIC MODEL –  Price per DVD Amount to Spend Number of DVDs bought –  12 dollars per DVD 55 dollars n DVDs 55 – 12n Problem Solving A2H CH 1 Equations and Inequalities MENU

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 ALGEBRAIC MODEL +  Price per minute Initial Fee Number of Minutes + .08 dollars per minute 5 dollars n Minutes n Problem Solving A2H CH 1 Equations and Inequalities MENU

Linear Inequalities A2H CH 1 Equations and Inequalities MENUAPPENDIX ABRIDGED ALGEBRA I INEQUALITY NOTES:

Linear Inequalities A2H CH 1 Equations and Inequalities MENUAPPENDIX Tougher inequality problems: Solve and graph:

Linear Inequalities A2H CH 1 Equations and Inequalities MENUAPPENDIX Tougher inequality problems: Solve and graph:

Linear Inequalities A2H CH 1 Equations and Inequalities MENUAPPENDIX Tougher inequality problems: Solve and graph:

Linear Inequalities A2H CH 1 Equations and Inequalities MENUAPPENDIX Tougher inequality problems: Solve and graph: 3-part inequalities must be “AND”

Linear Inequalities A2H CH 1 Equations and Inequalities MENUAPPENDIX Tougher inequality problems: What’s wrong with this:

Linear Inequalities A2H CH 1 Equations and Inequalities MENUAPPENDIX Tougher inequality problems: Solve and graph: No solution

Linear Inequalities A2H CH 1 Equations and Inequalities MENUAPPENDIX Tougher inequality problems: Solve and graph: ALL REAL NUMBERS

Absolute Value Equations & Inequalities A2H CH 1 Equations and Inequalities MENU ABRIDGED ALGEBRA I Absolute Value NOTES:

Absolute Value Equations & Inequalities A2H CH 1 Equations and Inequalities MENU Solve this absolute value inequality: NO SOLUTION

Absolute Value Equations & Inequalities A2H CH 1 Equations and Inequalities MENU Solve this absolute value inequality: ALL REAL NUMBERS

A2H CH 1 Equations and Inequalities MENU OPENERS ASSIGNMENTS EXTRA PROBLEMS APPENDIX

A2H CH 1 Equations and Inequalities MENUAPPENDIX Properties of equality Evaluate Absolute Value Inequality Literal Equations Ordering Numbers Simplify Solve (with fractions) Solve Absolute Value Inequality

Properties of addition and multiplication Identify the property illustrated: Inverse Property (addition) Associative property (addition) Identity property (addition) Identity property (multiplication) Inverse property (multiplication) Distributive Property A2H CH 1 Equations and Inequalities MENUAPPENDIX Closure property (addition)

Evaluate the following for the values x=4 and y=9 A2H CH 1 Equations and Inequalities MENUAPPENDIX

SOLVE FOR X HOME A2H CH 1 Equations and Inequalities MENUAPPENDIX

Write this equation in function form (solve for y) A2H CH 1 Equations and Inequalities MENUAPPENDIX

11. Write the following numbers in increasing order: HOME A2H CH 1 Equations and Inequalities MENUAPPENDIX

Simplify the following. A2H CH 1 Equations and Inequalities MENUAPPENDIX

SOLVE FOR X. A2H CH 1 Equations and Inequalities MENUAPPENDIX

SOLVE FOR X. A2H CH 1 Equations and Inequalities MENUAPPENDIX

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. The average of the extremesThe tolerance (difference between the average and extremes) HOME A2H CH 1 Equations and Inequalities MENUAPPENDIX

One of these inequalities is NO SOLUTION, the other is ALL REAL NUMBERS, can you identify which one is which? HOME A2H CH 1 Equations and Inequalities MENUAPPENDIX ALL REAL NUMBERS NO SOLUTION