Expressions, Equations, and Inequalities Chapter 1 Patterns and Expressions Sec. 1-1 Describe the pattern using words. Draw the next figure in the pattern. Identify a Pattern Definitions: a symbol usually a letter that represents one or more numbers. Numerical Expression: mathematical phrase that contains numbers and operation symbols. Algebraic Expression: mathematical phrase that contains one or more variables. Constants: values that do not change. Variable: These figures are made with toothpicks. Expressing a Pattern with Algebra A. How many toothpicks are in the 20th figure? Use a table of values with a process column to justify your answer. B. What expression describes the number of toothpicks in the nth figure? who new that toothpicks could be so much fun? What is the total cost if you buy the food plus 9 goldfish? Use a table to find the answer. You buy a bag of fish food and some goldfish. The graph shows the total cost depending on how many goldfish you buy. So you think you GOT IT? $19 ANSWER: 100 ANSWER: 5n ANSWER: Properties of Real Numbers Sec. 1-2 Algebraic Expressions Sec. 1-3 Solving Equations Sec. 1-4 Solving Inequalities Sec. 1-5 Absolute Value Equations and Inequalities Sec. 1-6 Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers Numbers you can write as a fraction.

Include decimals that terminate.

Include decimals that repeat. Have decimal representations that neither terminate nor repeat.

can not be written as a fraction. Variables: Quantities whose values change. Irrational #'s Rational #'s Integers Whole #'s Natural #'s Graph the real numbers -1/5, √2, 2.6 Graphing Real Numbers on the Number Line b) (3 * 4) * 5 = (4 * 3) * 5 a) 3(g + h) + 2g = (3g + 3h) + 2g Identify the Property Shown Mult. Addition Closure

Commutative

Associative

Identity

Inverse

Distributive Properties of Addition and Multiplication Using an inequality symbol ( < or > ), how do √85 and 8.9 compare? Subsets of the Real Numbers b) You had $150, but you are spending $2 each day. What algebraic expression models this situation? a) Write an algebraic experession models the word phase "one less than the product of six and w" ? EX 1: b) c3 - d/8; for c = 1/4 and d = 1 a) 2r + 5(s + 6) - 1; for r = 3 and s = -9 What is the value of the expression for the given values of the variables? EX 3: d) -(x + 4y) + 5(3x - y) c) 2a2 + 3b2 + 6b2 + 5a2 EX: Simplify by combining like terms. d) -2(y - 1) = -16 + y b) 12b = 18 c) -27 + 6y = 3(y - 3) a) x - 8 = -10 EX: Solve #1 RULE:

WHAT YOU DO TO ONE SIDE, YOU MUST DO TO THE OTHER. d) 2x - 12 + 3x = 2(2x - 6) + x c) 7x + 6 - 4x = 12 + 3x - 8 b) -x + 2(5x -1) = 2(3x + 4) +x a) 1 + 5x -6 = 6x - 5 - x Is the equation sometimes, always, or never true? 37 mi 44 mi 46 mi Essex Highville Dayton Newtown The map shows distances between towns in miles. You and your little friend drive together from Newtown to Essex. You drive first, for a total of 40 miles. Then your little friend drives for 1.5 hours, reaching Essex. What is your friends average rate? c) -n + 6 < 7n + 4 b) 7x +9 > 10x - 12 a) 5y - 8 < 12 EX: Solve i.e.: -3x < 12 Just like solving an equation, only difference is when mult. or div. by a negative, you MUST REVERSE the inequality. e) 6x + 9 < 3 or 3x - 8 > 13 d) -9 < t + 4 < 10 EX: Solve compound inequalities i.e. x > -2 and x < 5 x < -2 or x > 1 deff: Compound Inequalities: 2 inequalities joined by "and" or "or". 2x - 1 = 5 Solving an absolute value equation. Then graph the solution: Absolute Value:

the distance a number is from zero on the number line (distance is always postive). 3 x + 2 - 1 = 8 Extraneous Solutions:

solution derived from original equation that is not a solution to the original equation. x = -3 or x = -1 EX: Decide whether the given number is a solution to the equation. 3x+2 = 4x + 5 5x - 2 = 7x + 14 Solve the equation. 3x - 1 >≤ 5 2x + 4 <≥ 6 Land of Gor There are ALWAYS 2 set-ups EX: Rewrite the absolute value inequality as a compound inequality. Absolute value inequalities 4x + 3 < 5 5x + 10 > 15 3x - 4 <≤ 8 EX: Solve and Graph

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