Chapter 2

A LOT of time is spent in algebra learning how to solve equations and then solving them for various purposes. So, it goes without saying that we really need to understand what it means for something to “solve” an equation. First, let’s make sure we understand what an equation is: Definition of an Equation Exercise 1: Which of the following is not an equation? Equations can be either true, like (1) above, or false, like (4) above, depending on whether the two expression are equal (true) or not equal (false). a)Why can’t you determine whether this equation is true?

Solution(s) to an Equation A value for a variable is called a solution to the equation if, when substituted into both expressions, results in the equation being true. Exercise 3: Determine whether each of the following values for the given variable is a solution to the given equation.

You spent a lot of time in 8th grade Common Core Math solving linear equations (ones where the variable is raised to the first power only). In fact, the expectation is that you mastered solving linear equations. In today’s lesson, we will be solving linear equations where the variable only occurs once. We will solve these equations by seeing the structure of the expression involving x and using this structure to “undo” what has been done to it.

This is the most basic of all equation solving techniques. It is the most important solving technique in all of mathematics. Be clear on this: Solving Equations Using Inverse Operations If the variable you are solving for shows up only once, identify the operations that have been done on it and reverse them in the opposite order in which they occur. Now reverse them.

Exercise 3: Solve the following equation two different ways. Often equations can be solved in multiple ways. Let’s take a look at the next problem to see an example.

Exercise 4: Set up equations that translate the following verbal phrases into mathematics and then solve the equations. a)Ten less than five times a number results in thirty five. What is the number? b)When three times the sum of a number and seven is increased by ten, the result is four. What is the number?

The expectation of the Common Core is that students have mastered solving all types of linear equations in 8 th grade Common Core mathematics. In this lesson, we simply present a variety of linear equations for you to practice solving. Exercise 1: Solve each of the following “two-step” linear equations.

Exercise 2: Solve and check each of the following linear equations. Check:

Properties of Equality Additive Property of Equality Multiplicative Property of Equality The Common Core asks us not only to know the how but also the why. Generally, we justify the steps we take in solving linear equations using the commutative, associative, and distributive properties of real numbers along with the following two properties of equality.

O.k. That was a reasonably simple two-step equation. Now, let’s go for the full experience.

b)What do you think this tells you about the solutions to this equation? a)Justify each step in solving the equation.

b)Solve this equation by manipulating each side of the equation c)What does this final result tell you about this equation?

Although word problems can often be some of the most challenging for students, they give us great opportunities to refine our understanding of the relationships between quantities and how to manipulate expressions to solve equations. When you solve any real world problem in mathematics you are modeling a physical situation with mathematical tools, such as equations, diagrams, tables, as well as many others. MODELING AND SOLVING LINEAR WORD PROBLEMS 1.Clearly define the quantities involved with common sense variable using let statements. 2.Use your let statements to write out expressions for the quantities you are interested in. 3.Carefully translate these expressions into an equation. 4.Solve the equation. 5.Answer the question and check for the reasonableness of your answer. Exercise 1: The sum of a number and five more than the number is 17.

Exercise 2: The difference between twice a number and a number that is 5 more than it is 3. Find the number.

Exercise 3: Evie and her father are comparing their ages. At the current time, Evie’s father is 36 years older than her. Three years from now, Evie’s father will be five times her age at that point. How old is Evie now?

Exercise 4: Kirk has 12 dollars less than Jim. If Jim spends half of his money, and Kirk spends none, then Kirk will have two dollars more than Jim. How much money did they both start with?

Integers and Consecutive Integers Consecutive EvensConsecutive Odds Exercise 1 : Let’s work with just two consecutive integers first. Say we have two consecutive integers whose sum is eleven less than three times the smaller integer.

Exercise 2 : I’m thinking of three consecutive odd integers. When I add the larger two the result is nine less than three times the smallest of them. What are the three consecutive odd integers?

Exercise 3 : Three consecutive even integers have the property that when the difference between the first and twice the second is found, the result is eight more than the third. Find the three consecutive even integers.

Exercise 5: In an opera theater, sections of seating consisting of four rows are being laid out. It is planned so each row will be one more seats than the one before it and 126 people must be seated in each section. How many people will be in each row?

Exercise 1: Sydney’s parents pay her according to her grades. For each “A” that she earns, they add $12 and for each “C” they subtract $5 dollars. On a recent grade period, she got only “A’s” and “C’s” for her 6 classes. If she got a total of $21, how many of each grade did she get?

Exercise 2: Alex spent $3,100 more on his car than Bailee did. Elyse spent $675 less than Bailee spent on her car. Altogether, the three spent $31,501. How much did each spend on their cars?

Exercise 3: The perimeter of a rectangle is 76in. The length is two more than twice the width. Find the dimensions.

Exercise 3: Find three consecutive even integers such that the sum of twice the first and three times the third is fourteen more than four times the second.

At this point we should feel very competent solving linear equations. In many situations, we might even solve equations when there are no actual numbers given.

The rules for solving linear equations (and all equations) don’t depend on whether the constants in the problem are specified or not. The biggest difference in #1 between (a) and (b) is that in (b) you have to leave the results of the intermediate calculation undone.

Many times this technique is used when we want to rearrange a formula to solve for a quantity of interest. a.If a rectangle has a length of 12 inches and a width of 5 inches, what is the value of its perimeter? Include units.

So far we have concentrated on solving equations. Remember, all solving an equation consisted of was finding values of the variable that made the two expressions equal (in other words made the equation true). We can also judge the truth value of a statement that is in the form of an inequality. Exercise 1: For each inequality, state whether it is true or false. Inequality Definition

Exercise 3: Give the truth values for each of the following statements. Draw a number line to support your work.

Exercise 5: For each of the following inequalities, determine if it is true or false at the given value of the replacement variable.

Just as we can solve linear equations by using properties of expressions (commutative, associative, and distributive) and equations (addition and multiplication properties), we can do the same for inequalities. But, we have to make sure we know what those properties are. Let’s test them. a.If we add 3 to both sides of the inequality, what is the resulting inequality? Is it true? b.If we subtract 4 from both sides of the inequality, what is the resulting inequality? Is it true? c.If we multiply both sides of the inequality by 2, what is the resulting inequality? Is it true? d.If we divide both sides of the inequality by 2, what is the resulting inequality? Is it true?

Properties of Inequalities

a.Solve the inequality by applying the properties of inequalities. b.Write 5 numbers that make the finale solution true and plot them on the number line below. c.Now graph all of the solutions on the number line below (the Solution Set).

a.Solve the inequality by applying the properties of inequalities. b.Write 5 numbers that make the finale solution true and plot them on the number line below. c.Now graph all of the solutions on the number line below (the Solution Set).

a.Write the solutions using set notation. When solving inequalities it is important to be able to represent the solutions as a set of numbers using set notation. This notation is used to represent the set of all solutions of an equation or inequality b.Now graph all of the solutions on the number line below (the Solution Set).

a.Write the solutions using set notation. b.Now graph all of the solutions on the number line below (the Solution Set).

a.Write the solutions using set notation. b.Now graph all of the solutions on the number line below (the Solution Set).

a.Write the solutions using set notation. b.Now graph all of the solutions on the number line below (the Solution Set).

Linear inequalities tend to have an infinite amount of values that solve the inequality. Sometimes, we put two (or more) inequalities together and ask what x values make both true (AND) and which make either one or the other true (OR). You will deal with AND and OR along with truth values for the remainder of Algebra, so let’s discuss them in an exercise. Exercise 1: Consider each of the following compound inequality statements. Determine the truth value of both inequalities and then determine the overall truth value Truth Values for And and OR Inequalities 1.A compound AND statement will be true only if all of the individual statements are true. 2.A compound OR statement will be true if at least one of its individual statements is true.

Exercise 4: On the number lines below, shade in all values of x that solve the compound inequality. List some values:

Single Inequality:________________________

Inequalities involving AND are almost always universally written as a single inequality because these tend to show us how all values of x are between two numbers. Exercise 2: Graph each inequality on the number line, then rewrite each as two inequalities including the AND connector. Two Inequalities:________________________ Exercise 3: For each of the following graphs, write a compound inequality that describes all of the numbers shown graphed. Compound Inequality:___________________

a.Solve the compound inequality. b.Write the solutions using set notation. c.Graph the solution set on the number line.

a.Solve the compound inequality. b.Write the solutions using set notation. c.Graph the solution set on the number line.

a.Solve the compound inequality. b.Write the solutions using set notation. c.Graph the solution set on the number line.

a.Solve the compound inequality. b.Write the solutions using set notation. c.Graph the solution set on the number line.

We will often want to talk about continuous segments of the real number line. We’ve already done work with this in the last lesson using what is known as inequality or set-builder notation. Today we will see a very simple way of showing these segments using interval notation. Interval Notation Exercise 1: For each, graph the inequality and write the solution using interval notation. Interval Notation:___________________

Exercise 3: Two inequalities have solution sets given in interval notation below. a.Write an interval that represents all values that are solutions to both inequalities (AND). b.Write an interval that represents all values that are solutions to either of the inequalities (OR).

Exercise 1: A school is taking a field trip with 195 students and 10 adults. Each bus can hold at most 40 students. We need to determine the smallest number of busses needed for the trip. Exercise 2: Find all numbers for which five less than half the number is at least seven.

Exercise 3: Find all numbers such that twice the sum of the number and eight is at most four. Solve this problem by setting up and solving an inequality.