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

Example 1: Find the value of each expression.

Example 2: Evaluate the expression for

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, …}

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:

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

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

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

Example 2: Simplify each expression.

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

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

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.

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.

Example 4: Solve each equation for the specified variable.

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.

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.

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”

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

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.

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

Example 1: Solve and graph each inequality.