Indicating the solution to an inequality such as [latex]x\ge 4[/latex] can be achieved in several ways.

We can use a number line as shown below. The blue ray begins at [latex]x=4[/latex] and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.

We can use set-builder notation: [latex]\{x|x\ge 4\}[/latex], which translates to “all real numbers x such that x is greater than or equal to 4.” Notice that braces are used to indicate a set.

The third method is interval notation, where solution sets are indicated with parentheses or brackets. The solutions to [latex]x\ge 4[/latex] are represented as [latex]\left[4,\infty \right)[/latex]. This is perhaps the most useful method as it applies to concepts studied later in this course and to other higher-level math courses.

The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval, or a set of numbers in which a solution falls, are [latex]\left[-2,6\right)[/latex], or all numbers between [latex]-2[/latex] and [latex]6[/latex], including [latex]-2[/latex], but not including [latex]6[/latex]; [latex]\left(-1,0\right)[/latex], all real numbers between, but not including [latex]-1[/latex] and [latex]0[/latex]; and [latex]\left(-\infty ,1\right][/latex], all real numbers less than and including [latex]1[/latex]. The table below outlines the possibilities.

try it

We are going to look at a line with endpoints along the x-axis.

1. First we will adjust the left endpoint to (-15,0), and the right endpoint to (5,0)
2. Write an inequality that represents the line you created.

3. If we were to slide the left endpoint to (2,0), what do you think will happen to the line?

4. Now what if we were to slide the right endpoint to (11,0), what do you think will happen to the line? Sketch on a piece of paper what you think this new inequality graph will look like.

Show Solution

With endpoints (-15,0) and (5,0), the values for x on the line are between -15 and 5, so we can write [latex]-15<x<5[/latex]. We made it a strict inequality because the dots on the endpoints of the lines are open.

Moving the left endpoint towards the right endpoint shortens the line. Then moving the right endpoint away from the left endpoint lengthens the line again.

think about it

In the previous examples you were given an inequality or a description of one with words and asked to draw the corresponding graph and write the interval. In this example you are given an interval and asked to write the inequality and draw the graph.

Given [latex]\left(-\infty,10\right)[/latex], write the associated inequality and draw the graph.

In the box below, write down whether you think it will be easier to draw the graph first or write the inequality first.

Show Solution

We will draw the graph first.

The interval reads “all real numbers less than 10,” so we will start by placing an open dot on 10 and drawing a line to the left with an arrow indicating the solution continues to negative infinity.

To write the inequality, we will use < since the parentheses indicate that 10 is not included. [latex]x<10[/latex]

In the following video, you will see examples of how to write inequalities in the three ways presented here: as an inequality, in interval notation, and with a graph.

Using the Properties of Inequalities

When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equations. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number, we must reverse the inequality symbol.

Watch the following two videos for a demonstration of using the addition and multiplication properties to solve inequalities.

Solving Inequalities in One Variable Algebraically

As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.