2.7 Solve linear inequalities

Graph inequalities on the number line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

What about the solution of an inequality? What number would make the inequality $x>3$ true? Are you thinking, ‘ x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality $x>3$ .

We show the solutions to the inequality $x>3$ on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of $x>3$ is shown in [link] . Please note that the following convention is used: light blue arrows point in the positive direction and dark blue arrows point in the negative direction.

The graph of the inequality $x\ge 3$ is very much like the graph of $x>3$ , but now we need to show that 3 is a solution, too. We do that by putting a bracket at $x=3$ , as shown in [link] .

Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.

We can also represent inequalities using interval notation. As we saw above, the inequality $x>3$ means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation , we express $x>3$ as $\left(3,\infty \right).$ The symbol $\infty$ is read as ‘infinity’. It is not an actual number. [link] shows both the number line and the interval notation.