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Chapter 1: Solving Equations and Inequalities This chapter is going to review important concepts from Algebra 1. 1-1 Expressions and Formulas Objective: students will be able to use the order of operations to evaluate expressions

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Example 1: Find the value of each expression.

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Example 2: Evaluate the expression for

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1-2 Properties of Real Numbers Objectives: students will be able 1) to classify real numbers and 2) use the properties of real numbers to evaluate expressions Real numbers: all the numbers we use in everyday life Natural numbers: {1, 2, 3, …} Whole numbers: {0, 1, 2, …} Integers: {…-2, -1, 0, 1, 2, …}

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Natural numbers, whole numbers, and integers are all included in the set of rational numbers. – Includes all integers, as well as all terminating or repeating decimals – Examples of rational numbers:

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Irrational numbers: nonterminating, nonrepeating decimals – Examples of irrational numbers:

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Example 1: Name the sets of numbers to which each number belongs.

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The properties of real numbers can be used to simplify algebraic expressions.

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Example 2: Simplify each expression.

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1-3 Solving Equations Objectives: students will be able to 1) translate verbal expressions into algebraic expressions and equations, and vice versa and 2) solve equations using the properties of equality

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Example 1: Write an algebraic expression to represent each verbal expression. a)7 less than a number b)Three times the square of a number c)The cube of a number increased by 4 times the same number d)Twice the sum of a number and 5

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Example 2: Write a verbal sentence to represent each equation. The difference of a number and 8 is -9. A number divided by 6 is equal to the number squared.

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Remember, when solving equations the goal is to get all variable terms on one side of the equation, and all constants on the other side. Example 3: Solve each equation.

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Example 4: Solve each equation for the specified variable.

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1-5 Solving Inequalities Objective: students will be able to solve inequalities What is the difference between solving an equation and solving an inequality? – When multiplying or dividing BOTH sides of an inequality by a negative number, the inequality sign must be reversed.

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When graphing inequalities on a number line: When an inequality is solved, if the variable is on the left the inequality symbol will tell you which way to shade. – For example, x < 5 will result in an open circle on 5 and then will be shaded to the left (since the arrow is pointing left). This only works when the variable is on the left hand side.

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There are two different types of notation you may be asked to use when writing your solution: set-builder notation or interval notation. Set-builder notation This is read as “the set of all numbers x such that x is less than 5”

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Interval notation The left number indicates the left bound of the graph, while the right number indicates the right bound.

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Parenthesis are used to indicate – 1) a graph is unbounded in a certain direction ( is unbounded left) is unbounded right – 2) a graph cannot equal a number, meaning that the graph contains an open circle Brackets are used to indicate closed circles.

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Let’s practice interval notation. Write each solution using interval notation.

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Example 1: Solve and graph each inequality.

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