Note: Solving General Inequalities can be found here. Graphing inequalities with two variables on a coordinate system can be found here in the Coordinate System and Graphing Lines, including Inequalities section. Solving Absolute Value Inequalities can be found here in the Solving Absolute Value Equations and Inequalities section. Linear Inequality Word Problems can be found here in the Algebra Word Problems section.
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 constants 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” (for example, you must be at least 6 to ride a certain ride), or “at most” (for example, 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 parenthesis) 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 (and hard bracket) as compared to the open circle (and soft bracket or parenthesis).
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; always use soft brackets for $ -\infty $ and $ \infty $. If you have to “skip over” any intervals or numbers, you do so by using the “$ \bigcup $” sign, which means union. 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. This is because 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.
| Inequality | Number Line Graph |
| $ 3x-8<\,\,7$ $ \displaystyle \begin{array}{l}\underline{{\,\,\,\,\,\,\,\,\,\,+8<+8}}\\\,\,3x\,\,\,\,\,\,\,\,\,\,<\,\,\,15\end{array}$ $ \displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\frac{{3x}}{3}\,\,\,\,\,\,\,\,<\,\,\frac{{15}}{3}$ $ \displaystyle \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,x<\,\,\,5\\\,\,\,\,\,\,\,\,\,(-\infty ,5)\end{array}$ |  Check with 0 (it’s red) $ \displaystyle 3(0)-8<7$ $ \surd $ |
| $ \displaystyle -4m+13\le 29$ $ \displaystyle \begin{array}{c}\underline{{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-13\le -13}}\\\,\,-4m\,\,\,\,\,\,\,\,\,\,\,\le 16\end{array}$ $ \displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{-4m}}{{-4}}\,\ge \frac{{16}}{{-4}}\text{ }\,\left( {\text{notice} \le \text{changes to} \ge } \right)$ $ \displaystyle \begin{array}{c}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m\,\,\ge \,\,-4\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {-4,\infty } \right)\end{array}$ |  Check with 0 (it’s red) $ \displaystyle -4(0)+13\le 29$ $ \surd $ |
Compound Inequalities
Compound inequalities combine more than one inequality to get a solution; there are 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”. |
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| $ 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> -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$. | 
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On to Coordinate Systems and Graphing Lines including Inequalities – you are ready!