Inequalities can include any of the following: ,,,, or .

The graphing of linear inequalities is knowing how to do two things: 1) How to graph lines (Which you already know how to do), and 2) On which side of the boundary lines to shade.

If the problem is solved for y, then you will shade less than problems below the boundary line and greater than problems above the boundary line. If the problem is not solved for y, then first solve it for y.

We will use a solid boundary line when the line is to be included (,) and a dashed line when the line is not to be included (,).

Example Problem.
 
 

Graph y<x+4	Given Problem
Boundary Line is y=x+4	Recognize the boundary line
Make Boundary Line dashed	Boundary Line does not include line. (<)
	Graph boundary line (dashed)
	Put shading below, because of the (<) inequality.

Solving Inequalities

Definition:

With Absolute value, what we have is an ambidextrous statement. As the "troubling statement" above points out, there are two possible solution to an absolute value statement. Therefore, we will have to find two solutions to these problems, before we can find a combined, final solution.
 

Compound Sentences

Jack tells you that he received a 'B' on his test. What percentage did he have on the test? You don't know for sure, but you do know that it must be greater than 86 and it must be less than 91. This is what we call in mathematics, a compound sentence. In mathematical notation this can be written as: 86<x<91. This is also an example of what is meant by between in mathematics. The mathematical statement: 86<x<91 could be stated in English, "x is between 86 and 91." What type of interval would this be and how would it be graphed?

Math Statement: Type of Math Statement: Compound Statement Conjunction:  AND Graph:  Type of Interval:  Open Interval This problem dealt with a distance which is less than or equal to a certain amount. Absolute value problems are basically distance problems. From this problem we should observe that distance problems of a less than type will use the conjuction AND and will generally result in a "sandwich" or "between" graph.

Sarah is at least 3 miles from her teammate Kathryn in a marathon race. If Kathryn is at mile 4 1/2, where is Sarah? We don't know if Sarah is ahead or behind. Sarah could be at mile 1 1/2 (or earlier) or she could be at mile 7 1/2 (or later). Mathematically we could say the following:

Math Statement: Type of Math Statement: Compound Statement Conjunction: OR Graph: Type of Interval: Not an Interval This problem dealt with a distance which is greater than or equal to a certain amount. Absolute value problems are basically distance problems. From this problem we should observe that distance problems of a greater than type will use the conjuction OR and will generally result in a "split" graph.

Problem #37

Part 1 of Problem	AND	Part 2 of Problem
	"left side" Absolute Value sign simply dropped.		"right side" Use opposite Inequality, and additive inverse of non-absolute value side of problem.
	Add 7 to both sides		Add 7 to both sides
	Divide both sides by 5		Divide both sides by 5

Final Answer:   AND

AND used because this was a less than distance problem