6.05 Graph inequalities with rational numbers
Introduction
Previously, we have learned how to  identify and write inequalities . In the previous year, we have also learned how to  graph inequalities involving integers.
This time, we will graph inequalities with rational numbers including fractions and decimals.
Graph inequalities with rational numbers
Remember that rational numbers can be represented on a number line, stretching off towards positive infinity on the right, and negative infinity on the left.
If we have x = \dfrac{19}{5}, we can change the improper fraction into a mixed number or to a decimal number, to make it easier to locate on the number line.. If we know that \dfrac{19}{5}=3 \dfrac{4}{5}=3.8 we can plot the value of x=3 \dfrac{4}{5} or x=3.8 as follows:
When we say "x is less than or equal to \dfrac{19}{5}", we're not just talking about one number. We can draw a ray (a directed line) to represent all of points including fractions and whole numbers that are less than or equal to \dfrac{19}{5}, since all of them are included in the inequality.
The plot of x \leq \dfrac{19}{5} should look like the following:
What if we instead want to plot the very similar inequality x<\dfrac{19}{5}? The only difference now is that x cannot be equal to \dfrac{19}{5}, and so the plot should not include the point where x = \dfrac{19}{5}.
So we want to plot the same ray, but leave off the point at the end where x = \dfrac{19}{5}. To represent this we draw the plot with an unfilled circle, instead of a filled in circle, to show that \dfrac{19}{5} is not included:
The plot of x \lt \dfrac{19}{5} should look like the following:
To plot a greater than or greater than or equal to inequality, we instead want to show all of the numbers with larger value than a particular number. This is as easy as drawing a ray in the other direction instead, pointing to the right off towards positive infinity.
For example, the inequalities x \geq \dfrac{19}{5}and x > \dfrac{19}{5} are plotted below:
A plot of x \geq \dfrac{19}{5}
A plot of x > \dfrac{19}{5}
To plot inequalities on the number line:
- For \leq \text{ and } \geq we use a filled in or closed dot to start the ray.
- For < \text{ and } >, we use a hollow or open dot to start the ray.
- To check your ray is going the right way, choose a value which makes the inequality true and make sure your ray covers it.
Examples
Example 1
Plot the inequality x<\dfrac{5}{3}.
Example 2
State the inequality for x that is represented on the number line as a decimal.
To plot inequalities on the number line:
- For \leq \text{ and } \geq we use a filled in or closed dot to start the ray.
- For < \text{ and } >, we use a hollow or open dot to start the ray.
- To check your ray is going the right way, choose a value which makes the inequality true and make sure your ray covers it.