Ordering Positive and Negative Numbers

We can use the number line to compare and order positive and negative numbers. Going from left to right, numbers increase in value. Going from right to left, numbers decrease in value. See the figure below.

This figure is a number line. Above the number line there is an arrow pointing to the right labeled increasing. Below the number line there is an arrow pointing to the left labeled decreasing.

Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers. Remember that we use the notation \(ais less than \(b\)) when \(a\) is to the left of \(b\) on the number line. We write \(a>b\) (read \(a\) is greater than \(b\)) when \(a\) is to the right of \(b\) on the number line. This is shown for the numbers \(3\) and \(5\) in the figure below.

This figure is a number line with points 3 and 5 labeled with dots. Below the number line is the statements 3 is less than 5 and 5 is greater than 3.

The number \(3\) is to the left of \(5\) on the number line. So \(3\) is less than \(5,\) and \(5\) is greater than \(3.\)

The numbers lines to follow show a few more examples.

This figure is a number line with points 1 and 4 labeled with dots.

\(4\) is to the right of \(1\) on the number line, so \(4>1.\)

\(1\) is to the left of \(4\) on the number line, so \(1<4.\)

This figure is a number line with points negative 2 and 1 labeled with dots.

\(-2\) is to the left of \(1\) on the number line, so \(-2<1.\)

\(1\) is to the right of \(-2\) on the number line, so \(1>-2.\)

This figure is a number line with points negative 3 and negative 1 labeled with dots.

\(-1\) is to the right of \(-3\) on the number line, so \(-1>-3.\)

\(-3\) is to the left of \(-1\) on the number line, so \(-3<-1.\)

Example

Order each of the following pairs of numbers using \(<\) or \(\text{>:}\)

\(\phantom{\rule{0.2em}{0ex}}14\_\_\_6\phantom{\rule{0.8em}{0ex}}\)
\(\phantom{\rule{0.2em}{0ex}}-1\_\_\_9\phantom{\rule{0.8em}{0ex}}\)
\(\phantom{\rule{0.2em}{0ex}}-1\_\_\_-4\phantom{\rule{0.8em}{0ex}}\)
\(\phantom{\rule{0.2em}{0ex}}2\_\_\_-20\)

Solution

Begin by plotting the numbers on a number line as shown in the figure below.

This figure is a number line with points negative 20, negative 4, negative 1, 2, 6, 9, and 14 labeled with dots.

Compare 14 and 6.	\(14\_\_\_6\)
14 is to the right of 6 on the number line.	\(14>6\)

Compare −1 and 9.	\(-1\_\_\_9\)
−1 is to the left of 9 on the number line.	\(-1<9\)

Compare −1 and −4.	\(-1\_\_\_-4\)
−1 is to the right of −4 on the number line.	\(-1>-4\)

Compare 2 and −20.	\(-2\_\_\_-20\)
2 is to the right of −20 on the number line.	\(2>-20\)

Ordering Positive and Negative Numbers