Algebra: Graphing on Number Lines

On this page we hope to clear up problems that you might have with graphing on number lines. Number lines aren't used very extensively, but they can be useful when you need a graph, but not of an equation.

Read on or follow the links below to start understanding how to graph on a number line!

Inequalities
Conjunctions
Domain
Quiz on graphing on number lines

Inequalities

In this section we will help you understand how to graph inequalities on number lines.

Many times you will have a statement such as x > 5 that needs to be graphed. Because this is not an equation, it does not need to be graphed on the coordinate plane. A number line does the job just fine!

Some conventions that need to be remembered when graphing on a number line are explained below.

1. An open circle is placed on the number line to show that the number denoted at the circle is not included in the solution set.
2. A circle that is filled in is placed on the number line to show that the number denoted at the circle is included in the solution set.

Example:

1. Graph:    x < 4
   Solution: The problem asks you to graph all numbers that
             are less than 4.

             The graph: Example Graph

Conjunctions or Complex Inequalities

The word conjunction means there are two conditions in a statement that must be met. Therefore, a mathematical statement such as the following is a conjunction: 5 < x < 10. There are two things to remember when dealing with conjunctions. They are outlined below.

1. The greater than or less than signs will always be pointing in the same direction (i.e., you will never see the following: 7 > x < 2).
2. Look out for statement that cannot be true, such as the following: 10 < x < 5.

Example:

1. Graph:    -2 < x <= 4
   Solution: The problem, which is a conjunction, asks for
             a graph of all the numbers between -2 and 4.  Be
             sure to note that -2 is not included in the
             solution, while 4 is.

             The graph: Example Graph

Domain

In all of math, and even in non-math subjects that utilize the things you learn in math (such as physics and chemistry), restrictions are typically placed on the problem either by the wording of the problem or by the nature of the problem. For example, if a father sends his daughter on an errand to the local market to buy eggs, and gives her 90 cents, and eggs cost 20 cents a piece, the total number of eggs she can buy and the total amount of money she can spend is restricted. This can easily be represented by an equation -- (TC) = 20(NE)

TC (total cost) cannot exceed 90 cents. With that in mind, you realize the girl can only buy up to 4 eggs. It is possible to go and decide not to buy eggs (although this wouldn't be very smart if you actually needed eggs), so you could use 0 for NE. However, 1 might make a little more sense! We already established the fact that you could buy 4, but 2 and 3 are also possible. Anyone who tried to buy half an egg or only a quarter of an egg would probably have an easier time with life if they lived in a white padded cell, so you can only use whole numbers. Also, it wouldn't make sense to try and by a negative number of eggs, so you can only use {0, 1, 2, 3, 4} for possible solutions to the problem.

The set of numbers you can use to solve an equation is called the domain. All equations and inequalities have domains.

Examples:

1. Graph:    x < 3,  D = {Integers}
   Solution: This problem asks for a graph of all integers
             less than three.

             The graph: Example Graph

2. Graph:    x < -1,  D = {Positive Integers}
   Solution: The solution is the null set because there are
             no positive integers that are less than
             -1.

Take the quiz on graphing on number lines. The quiz is very useful for either review or to see if you've really got the topic down.

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