Graphing intervals on the real line

Section 1.5 Graphing intervals on the real line

We can represent intervals of real numbers graphically on the real line by shading in the relevant portions. A filled-in circle indicates that an endpoint is included, an empty circle indicates that the endpoint is not included, and an arrow indicates that the interval extends forever in that direction. When an interval is expressed in the form \(\{x : x>a\}\text{,}\) \(\{x : x\lt a\}\text{,}\) \(\{x : x\geq a\}\text{,}\) or \(\{x : x\leq a\}\text{,}\) this is straightforward.

Example 1.5.1. Graphing Intervals.

The interval \((-\infty,-3]=\{x: x \leq -3\}\text{,}\) is represented by the following graph.

The drawing of the interval from negative infinity to negative three on the number line. — Figure 1.5.2.

The filled-in circle at the point \(x=-3\) indicates that \(-3\) is included in the interval. The arrow indicates that all real numbers less than \(-3\) are also included.

The interval \(\{ x: x > 2 \}\) is represented by the following graph.

The empty circle at the point \(x=2\) indicates that \(2\) is not included in the interval, and the arrow indicates that all real numbers greater than \(2\) are included.

Dividing by a negative number. You may recall that when dividing both sides of an inequality by a negative number, the inequality switches direction. This is because dividing by a negative number corresponds to subtraction, as for example in:

\begin{equation*} \begin{aligned} -x &> 2 \\ -x+x &> 2 + x \\ 0 &> 2+x \\ 0 - 2 &> -2+2+x \\ -2 &> x \end{aligned} \end{equation*}

Thus, \(-x>2\) is equivalent to \(x\lt -2\text{.}\)

Graphically, the interval \(\{x : -x > 2\}\) is the mirror image of the interval \(\{x:x>-2\}\) on the number line.

Example 1.5.3. Dividing and Inequalities.

Represent \(\{x: -3x \lt 9\}\) graphically.

Solution.

In this example, the inequality is not yet expressed in the form \(x\lt a\) or \(x>a\text{,}\) and so we need first to rewrite the inequality. Dividing both sides by \(-3\text{,}\) remembering that this switches the direction of the inequality, gives us \(\{x : x > -3\}\text{:}\)

Example 1.5.4. Representing Unions and Intersections Graphically.

The set \(( - \infty, 2 ) \cup \big[ 4, \infty )\) is the union of two intervals, both of which we can represent graphically. The union is just represented by drawing both:

The set \(( - \infty, 1 ) \cap \big[ 0, \infty )\) is the intersection of two intervals. To represent this set graphically, we draw only the portion that is common to both: