2.9.2 Graphing Inequalities

Graphing Linear Inequalities

What number would make the inequality $x>3$ true? Are you thinking, “$x$ could be 4”? That’s correct, but $x$ could be 6, too, or 37, or even 3.001. Any number greater than 3 is a solution to the inequality $x>3$. We show all 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 circle at 3.

We can also represent inequalities using interval notation. There is no upper end to the solution to this inequality. In interval notation, we express $x>3$ as $(3,\infty )$. The symbol ∞ is read as “infinity.” It is not an actual number.

The figure below shows both the number line and the interval notation.

                                                                        $x>3$

$(3,\infty )$

The inequality $x\le 1$ means all numbers less than or equal to 1. Here we need to show that 1 is a solution, too. We do that by putting a closed circle at $x=1$. We then shade in all the numbers to the left of 1, to show that all numbers less than 1 are solutions. See the figure below.

                                                   $x\le 1$

$(-\infty ,1]$

There is no lower end to those numbers. We write $x\le 1$ in interval notation as $(-\infty ,1]$. The symbol −$\infty$ is read as “negative infinity.” The figure below shows both the number line and interval notation.

Inequalities, Number Lines, and Interval Notation

Try it

Try It: Graphing Linear Inequalities

For questions 1 - 2 use the following inequality:

$x>-3$

1.

Write the inequality on a number line.

Compare your answer:

2.

Write the inequality in interval notation.

Compare your answer: $(-3,\infty )$

For questions 3 and 4 use the following inequality:

$x\le 5$

3.

Write the inequality in interval notation.

Compare your answer: $(\infty ,5]$

4.

Write the inequality on a number line.

Compare your answer: