4.07 Inequalities on a number line

Lesson

Graphing inequality statements on the number line

Inequalities that include a variable can be represented nicely on a number line. Let's quickly recap plotting points on a number line.

Remember that all the real numbers can be represented on an infinite line called the number line, stretching off towards positive infinity on the right, and negative infinity on the left. Numbers further to the left are lesser numbers and numbers further to the right are greater numbers.

On the number line above, the integers are marked (and every fifth number is labeled). However, in between each whole number lies an infinite stream of rational and irrational numbers.

We can plot any real number we like on the number line. For example, if we know that $x=6$x=6, we can plot the value of $x$x as follows:

A plot of $x=6$x=6.

Now, what if we wanted to plot an inequality, such as $x\le4$x4?

When we say "$x$x is less than or equal to $4$4", we're not just talking about one number. We're talking about a whole set of numbers, including $x=4$x=4, $x=2$x=2, $x=0$x=0, $x=-1$x=1 and $x=-1000$x=1000. All of these numbers are less than or equal to $4$4.

If we plot all of the integers that are less than or equal to $4$4 on a number line, we get something that looks like this:

A first attempt at plotting $x\le4$x4.

So far so good. But what about fractions like $x=\frac{1}{2}$x=12? These numbers are also less than or equal to $4$4, so surely they should be shown on the plot too?

In fact, there are countless numbers that are less than or equal to $4$4, filling up all of the space in between each of the integers plotted above, continuing on to the left towards negative infinity. Rather than trying to plot all of these points (which would get messy quite quickly), we can draw a ray to represent all of these points, since all of them are included in the inequality.

The actual plot of $x\le4$x4.

What if we instead want to plot the very similar inequality $x<4$x<4? The only difference now is that $x$x cannot take the value of $4$4, and so the plot should not include the point where $x=4$x=4.

So we want to plot the same ray, but leave off the point at the end where $x=4$x=4. To represent this we draw the plot with a hollow circle, instead of a filled in circle, to show that $4$4 is not included:

A plot of $x<4$x<4.

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 $x\ge4$x4 and $x>4$x>4 are plotted below:

A plot of $x\ge4$x4.

A plot of $x>4$x>4.

Remember!

If the variable is written on the left of the inequality, then the arrow of the ray will always point in the same direction as the inequality symbol!

Practice questions

Question 1

Plot the inequality $x<0$x<0 on the number line below.

Question 2

Plot the inequality $x<2$x<2 on the number line below.

Question 3

State the inequality for $x$x that is represented on the number line.

Graphing solutions to inequalities

Now let's consider an inequality such as $x+3>5$x+3>5. What would we plot for this inequality?

As in the case of $x\le4$x4 above, what we want to plot on the number line are all of the possible values that the variable can take - that is, the solutions of the inequality. The inequality $x+3>5$x+3>5 has the solutions "all numbers which, when added to $3$3 result in a number greater than $5$5". This is a little bit of a mouthful already, and there are definitely much more complicated inequalities than this!

So in order to plot the solutions to an inequality such as $x+3>5$x+3>5, it will be easiest to first solve the inequality. In this case, we can subtract $3$3 from both sides to get $x>2$x>2. So the plot will show "all numbers greater than $2$2" on the number line, which looks like this:

Remember

When solving an inequality:

• Reversing the order of the inequality will reverse the inequality symbol too.

When plotting an inequality:

• The symbols $<$< and $>$> don't include the end point, which we show with a hollow circle.
• The symbols $\ge$ and $\le$ do include the endpoint, which we show with a filled circle.

Practice questions

Question 4

Consider the inequality $3+x<2$3+x<2.

1. Solve the inequality.

2. Now plot the solutions to the inequality $3+x<2$3+x<2 on the number line below.

Question 5

Consider the inequality $-88<x4. 1. Solve the inequality. 2. Now plot the solutions to the inequality$-88<x4 on the number line below.

Question 6

Consider the inequality $-4<-3+x$4<3+x.

1. Solve the inequality.

2. Now plot the solutions to the inequality $-4<-3+x$4<3+x on the number line below.

Outcomes

6.14b

Solve one-step linear inequalities in one variable, involving addition or subtraction, and graph the solution on a number line