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

Example 1: Find the value of each expression.

Example 2: Evaluate the expression for

1-2 Properties of Real NumbersObjectives: 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 EquationsObjectives: 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 numberb) Three times the square of a

numberc) The cube of a number increased

by 4 times the same numberd) 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.