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Introduction to Inequalities

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- What is an inequality?
- Plotting inequalities on a number line

When we have an equation such as x = 4 we have a specific value for our variable. With inequalities we will give a range of values for our variable. To do this we will not use equals, but one of the following symbols:

The above symbols are inequality symbols. They relate two quantities as not being equal to each other. Let's look at a definition of an inequality:

Inequality: a mathematical statement that two quantities are not equal in value.

If we have an expression such as x < 4, this means our variable can be any number smaller than 4 such as − 2, 0, 3, 3.9 or even 3.999999999 as long as it is smaller than 4. If we have an expression such as x − 2, this means our variable can be any number greater than or equal to −2, such as 5, 0, −1, −1.9999, or even −2.

Because we don’t have one set value for our variable, it is often useful to draw a picture of the solutions to the inequality on a number line. We will start from the value in the problem and bold the lower part of the number line if the variable is smaller than the number, and bold the upper part of the number line if the variable is larger. The value itself we will mark with brackets, either ) or ( for less than or greater than respectively, and ] or [ for less than or equal to or greater than or equal to respectively.

Once the graph is drawn we can quickly convert the graph into what is called interval notation. Interval notation gives two numbers, the first is the smallest value, the second is the largest value. If there is no largest value, we can use ∞ (infinity). If there is no smallest value, we can use − ∞ negative infinity. If we use either positive or negative infinity we will always use a curved bracket for that value.

Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html