Introduction to Graphing Inequalities

To graph an inequality you must first determine the solution to the problem as if it were an equality. This will give you the boundary point. In this video, we will look at the finer details of what to do to find and graph the solution set.

 

Let’s continue this idea by looking at a few more examples of graphing inequalities.

Graph:     $ \color{#2f6683}{x-7 \geq 3}$

1. Solve the inequality as if it was an equation:

\[\begin{align*}
x-7&=3\\
x+{^-7}&=3\\
x+{^-7}\color{#2f6683}{+7}&=3\color{#2f6683}{+7}\\
x&=10
\end{align*}\]

 
2. Use the value you got as the boundary point for the graph: $x\geq 10$

solving an inequality boundary point
 
3. Make a few points that make   $x\geq 10$ true

picking points for the inequality
 
4. Draw a line from the boundary point that will represent all possible solutions to the inequality.

solution to x - 7 is greater than or equal to 3

Remember the closed circle means that 10 is part of the solution set.

 
Graphing Inequalities

5. Pick a point from the solution set and check it in the original inequality,

such as $\color{#2f6683}{x=12}$

\[\begin{align*}
x-7&\geq3 \\
\color{#2f6683}{12}+{^-7}&\geq3\\
\color{#2f6683}{5}&\geq3\\
\end{align*}\]

Graph:     $ {\frac{n}{4}<12}$

1. Solve the inequality as if it was an equation:

\[\begin{align*}
\frac{n}{4}&<12\\ \\[1px] \color{#f15a23}{4*}\frac{n}{4}&<12\color{#f15a23}{*4}\\ \\[1px] \color{#f15a23}{n}&<\color{#f15a23}{48} \end{align*} \]

2. Use the value you got as the boundary point for the graph.
3. Mark a few points that make   $n<48$ true.

4. Draw a line from the boundary point that will represent all possible solutions to the inequality.

Graphing the inequality n is less than 48

Remember the open circle means that 48 is not part of the solution set.

5. Pick a point from the solution set and check it in the original inequality, such as $\color{#f15a23}{x=40}$

graphing inequalities problem checks
\[\begin{align*}
\frac{n}{4}&<12\\ \\[1px] \frac{40}{4}&<12\\ \\[1px] \color{#f15a23}{10}&<12 \end{align*}\]