Linear Inequalities


This section covers:

(If you are graphing inequalities with two variables on a coordinate system, go here.)

Linear Inequality Word Problems can be found here.

Basic Linear Inequalities

Linear Inequalities are just like equations but the two sides may not be equal. Instead of an equal sign, we use \(>\) (greater than), \(\ge \) (greater than or equal to), \(<\) (less than), or \(\le \) (less than or equal to). An inequality can also have a \(\ne \) (does not equal) sign, which simply means that that the variable can be anything but what you get for the answer.

Inequalities are basically solved the same way as equalities or equations, but we have to be a little more careful moving things like numbers and variables back and forth, since we’re not dealing with simply an equal sign. Typically, when you solve them, you have a range of answers (meaning more than one answer), so we usually graph them on the number line.

Inequalities are used in situations where something has to be “at least” (like you must be at least 6 to ride a certain ride), or “at most” (like you need to spend at most $50 in the store). We’ll see these words when we get to the word problem section.

Note that when we have \(<\) or \(>\) (without the equal), we use an open circle (or sometimes an open parentheses) when we’re graphing, and when have \(\le \) or \(\ge \), we have a closed circle (or sometimes a bracket) when we’re graphing (this is sometimes called inclusive). The way I remember this is when there is more pencil used to create the \(\ge \) sign instead of the \(>\) sign, for example, there is more pencil used to create the closed circle as compared to the open circle.

You must have the variable on the left hand side when you use these rules to know where to graph:

  • When you have the less than sign (\(<\)), you want to graph to the left (both start with “le”); when you have the greater than sign (\(>\)) you graph to the right.
  • You can also graph in the direction where the inequality sign is pointed.
  • You can also plug in a number to see if it works: if it works, it should be on the colored part of the number line – like 3 is for the inequality “\(x<4\)” (since 3 is less than 4).

Here are some examples, including the solutions in Interval Notation. Remember that with Interval Notation, you always go from lowest to highest number with “(“ (soft brackets) if the inequality doesn’t hit the point, and “[“ (hard brackets) if the inequality does hit the line.

If you have to skip over any numbers, you do so by using the “\(\bigcup \)” sign, which means union, or put the things together. Also note that if we just have to graph one number, we just put a circle on it.

Inequality Notation Interval Notation Number Line Graph
\(x<4\) \((-\infty ,4)\)
\(x\ge -2\) \(\left[ {-2,\infty } \right)\)
\(x=-3\) \(\{-3\}\)
\(x\ne 1\) \(\left( {-\infty ,1} \right)\cup \left( {1,\infty } \right)\)

Note that when you solve an inequality, you pretend like the inequality is an equal sign.  The only thing you have to worry about is multiplying or dividing by a negative number; when you do this, you have to reverse the inequality symbol (change \(<\) to \(>\), or \(>\) to \(<\)). You can see an example of this in the last equation below. The reason we need to do this, is when we have a negative coefficient (what comes before the variable), we’d eventually have to move the variable with its coefficient to the other side and make it positive, so it would be on the other end of the inequality sign.

At this point, we are not multiplying or dividing by variables (since we don’t know the signs of them); we will learn how to do that later.

Inequality Number Line Graph


      \(\begin{array}{l}\underline{{\,\,\,\,\,\,\,\,+8<+8}}\\3x\,\,\,\,\,\,\,\,\,<\,\,\,15\\\,\,\,\,\,\,\frac{{3x}}{3}\,\,\,<\,\,\frac{{15}}{3}\\\,\,\,\,\,\,\,\,\,\,\,\,x<\,\,\,5\\\,\,\,\,\,\,\,\,\,(-\infty ,5)\end{array}\)

Check with 0 (it’s red)

\(\displaystyle 3(0)-8<7\)

It works!

\(\displaystyle -4m+13\le 29\)

\(\displaystyle \begin{array}{c}\,\,\,\,\,\underline{{\,\,\,\,\,\,\,\,\,\,\,\,-13\le -13}}\\-4m\,\,\,\,\,\,\,\,\,\,\,\,\,\le 16\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{-4m}}{{-4}}\,\,\ge \frac{{16}}{{-4}}\text{ }\\\,\,\,\,\,\,\,\,\,\,\left( {\text{notice} \le \text{changes to} \ge } \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m\,\ge -4\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {-4,\infty } \right)\end{array}\)

Check with 0 (it’s red)

\(\displaystyle -4(0)+13\le 29\)

It works!

Compound Inequalities

Compound inequalities combine more than one inequality to get a solution.  There are basically three different ways to write these, as shown below. Remember that “or” means either one can work, and “and” means that they both have to work.

You can also think of this:  “or” means that you only need one line when you graph the inequalities, “and” means you need both lines.

Here are some examples; notice that \(x<4\) and \(x\ge -2\) can be written as \(-2\,\le x<4\) (do you see why?):

Compound Inequality Number Line Graph

\(x<4\,\,\,\,\text{and}\,\,\,x\ge -2\)

\(\begin{array}{c}-2\le x<4\\\left[ {-2,4} \right)\end{array}\)

Solution has to contain double lines from above since it’s an “and”.

\(x\le 1\text{ }\,\,\text{or }\,\,\text{ }x>4\)

\(\left( {-\infty ,1} \right]\cup \left( {4,\infty } \right)\)

Solution just has to contain one line from above since it’s an “or”.

\(x<4\,\,\,\,\text{or}\,\,\,x\ge -2\)

\(\,\,\left( {-\infty ,\infty } \right)\)

Solution just has to contain one line from above, so we include the whole number line (All Reals).

\(x\le 1\text{ and }x>4\)

\(\emptyset \text{ (no solution)}\)

Solution has to contain double lines, and since we don’t have any, there is no solution (nothing on graph).

\(x\le 1\text{ and }x\le 4\)

\(\left( {-\infty ,1} \right]\)

Solution has to contain double lines, so we only include where the graph is \(x\le 1\).

\(x\ge -1\text{ or }x\ge 4\)

\(\left[ {-1,\infty } \right)\)

Solution just has to contain one line from above, so the solution is \(x\ge -1\).



Try these by putting in numbers from the number line (in the pink, and not in the pink) to make sure you understand it!

Note that we will address Linear Inequalities in Two Variables in the Coordinate System and Graphing Lines Including Inequalities section here.

Note that we will address Absolute Value and Inequalities in the Solving Absolute Value Equations and Inequalities section here.

Learn these rules, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Coordinate Systems and Graphing Lines including Inequalities – you are ready!