Solving Inequalities Using Multiplication and Division

Topics for Today:

Our discussion about solving inequalities moved to solving by using the multiplication and division properties of inequality.  We solve inequalities using the same steps and procedures as solving equations, but there is one notable exception.  For cases when we either multiply or divide both sides of our inequality by a negative number, we must switch the inequality sign to preserve the truth of the inequality.  To illustrate why this works, we did a little exploration with simple inequalities in class to help understand why the "truth" of an inequality changes.

Vocabulary: multiplication property of inequality, division property of inequality

We considered the following examples in class:

Sections Covered in Textbook:

3-3: Solving Inequalities Using Multiplication & Division
       (pages 146-151)

Resources & Tutorials:

1) What is the division property of inequality?
2) What is the multiplication property of inequality?
3)  Solving inequalities using multiplication and division
4) Virtual Nerd Page with more tutorials.

Solving Inequalities Using Addition and Subtraction

Topics for Today:

Now that we understand what inequalities and solutions to inequalities are, we can now move into solving them.  When solving simple inequalities using addition and subtraction, we basically follow the same steps we use for solving simple equations.  For these problems, we will use the addition and subtraction properties of inequality to "undo" operations performed on a variable with the goal of getting the variable by itself.  Checking solutions to inequalities may not always locate our mistakes since there are an infinite number of possible solutions we can use to check ourselves.  Students will be encouraged to try out multiple possible solutions when checking.

Vocabulary: equivalent inequalities, addition property of inequality, subtraction property of inequality.

Graphic Credit:
https://www.mathwarehouse.com/number-lines/graph-inequality-on-number-line.php#examples1

Sections Covered in Textbook:

3-2: Solving Inequalities Using Addition and Subtraction (pages 140-144)

Resources & Tutorials:

1) What is the addition property of inequality? 
2) How do you solve an inequality using subtraction?
3) How do you solve an inequality using addition?

    (This includes putting the solution in set notation, which we did not discuss in class.)

Inequalities and Their Graphs

Topics for Today:

We moved on to Chapter 3 today.  Our discussion has moved from equations where both sides are equal, to inequalities where one side is larger or smaller than the other.  We also discussed the difference between solutions where the endpoint is included vs excluded and explored the graphs of inequalities.  A solution to an inequality is any value that will make the inequality true.  Inequalities differ from equations because inequalities often have infinite solutions that are bound by a particular value whereas equations typically have a finite solution set.

Graphic Credit: https://www.katesmathlessons.com/graphing-inequalities.html

Vocabulary: inequality, solution to an inequality

Sections Covered in Textbook:

3-1: Inequalities and Their Graphs (pages 134-138)

Resources & Tutorials:

1) What is an Inequality?
2) How Do You Graph Inequalities?
    (This video also includes infinite sets which we did not discuss.)

Literal Equations

Topics for Today:

Today we continued our exploration of equations, but focused on literal equations and formulas.  Literal equations are just equations that have more than one variable.  Formulas are mathematical or scientific facts, rules, or relationships expressed with mathematical symbols.  Students have been using formulas for much of their mathematics studies, although they may not be aware.  Earlier in this chapter, we used the formula for distance (d=rt) and also perimeter of a rectangle {P=2(l+w)}.

We can use our knowledge of solving equations to move variables around in literal equations or formulas, to solve for a particular variable.  The properties of equality (addition, subtraction, multiplication, division and distributive) still apply here in these examples.

The first step in any solving equation problem it to identify the variable that you are solving for.  Once that is done, we follow the reverse order of operations to isolate the variable, and follow the same steps we used for solving multi-step equations.  As a reminder, here are those steps again:

Vocabulary: formula, literal equation

Sections Covered in Textbook:

2-6: Formulas (pages 111-114)

Resources & Tutorials:

1) What is a literal equation?
2) How do you solve a formula for a variable?
3) Summary of solving literal equations.

Equations and Problem Solving Part 2 - Uniform Motion

Topics for Today:

We continued our discussion about problem solving by investigating uniform motion problems.   Most students are familiar with the basic formula for motion: 
distance = rate * time, or in algebraic terms, d = rt.

Uniform motion problems fall into three main categories: same-direction travel, round-trip travel, or opposite-direction travel.  Depending upon what the problem is asking, we'll combine our problem data in different ways to find our answer, but in each case, we will still apply the general formula (d=rt) to set up our problem.

Drawing diagrams to help picture what is going on in the problem is another helpful strategy.  Using a table to solve problems helps organize all the supporting data, and provides a systematic way to solve more complex problems.  Students are encouraged to use a table and to draw a picture of what is going on in the question to help better understand what is being asked.

Sections Covered in Textbook:

2-5: Equations and Problem-Solving Part 2 (pages 103-110)

Resources & Tutorials:

1)  How to Solve Opposite-Directions problem.
2)  How to Solve Same-Directions problem.
3)  How to Solve Round-Trip Travel Problems.

Equations and Problem Solving Part 1

Topics for Today:

Now that we know how to solve all types of equations, we will use that knowledge to solve story problems.  There are many different types of story problems, but most of them can be categorized into one of several categories.  Today's lesson will focus on problem solving involving defining variables in terms of each other,  consecutive integers (which also involves defining variables in terms of another), and break-even problems.

Vocabulary: consecutive, break-even

Sections Covered in Textbook:

2-5: Equations and Problem Solving (pages 103 - 110)
We will continue working on this topic tomorrow with more examples.

Resources & Tutorials:

1) Solving Break-Even Problems
2) How to find the Break-Even Point
3) Solving Word Problems with Consecutive Integers

Solving Equations with Variables on Both Sides

Topics for Today:

Our discussion about solving equations moved on to situations where there are variables on both sides of an equation.  The basic properties of equality still apply when dealing with variable terms.  We used the addition, subtraction, multiplication, and division properties of equality to get variables on one side, and constants on the other side of the equation.  For equations with variables on both sides, it is possible for the equation to have infinitely many solutions (identity) or no solution at all.  Students will need to be on the lookout for these special cases, which show themselves is interesting mathematical ways.

Sections Covered in Textbook:

2-4: Equations with Variables on Both Sides (pages 96-100)

Resources & Tutorials:

1) Solving Equations with Variables on Both Sides
2) Solving Equations with Variables on Both Sides and Grouping Symbols