x<2	"x \text{ is less than } 2"
x>-5	"x \text{ is greater than } -5"
2x\leq-4	"\text{2 groups of } x \text{ is less than or equal to } -4"
x-3\geq17	"\text{3 less than } x \text{ is greater than or equal to } 17"

Inequalities that include a variable, such as the examples above, 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 smaller numbers and numbers further to the right are larger numbers.

We can plot any real number we like on the number line. For example, if we know that x=6, we can plot the value of x as with a solid dot as shown.

Similarly, if we know that x = \dfrac{19}{5}, we can plot the value of x=3 \dfrac{4}{5} as shown.

Ideas

Ineqeualities on the number line

Ineqeualities on the number line

Now, what if we wanted to plot an inequality, such as x \leq 4?

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

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

So far so good. But what about fractions like x = \dfrac{1}{2}, or irrational numbers like x = \sqrt{2}?

These numbers are also less than or 4, so surely they should be shown on the plot too?

To show all of the values less than or equal to 4, we can draw a ray (a directed line) to represent all of these points, since all of them are included in the inequality.

Shown is the plot of x \leq 4.

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

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

Shown is the plot of x \lt 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.

Shown is the plot of x \geq 4.

Shown is the plot of x > 4.

Here are two more examples of inequalities plotted on a number line:

A plot of x > -\dfrac{11}{20}

A plot of x \leq -20.7

Examples

Example 1

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

Worked Solution

Create a strategy

Use the fact that the circle is hollow and the ray is pointing to the right.

Apply the idea

The ray extends to the right of a hollow dot at point 1. It means that 1 is not included and any value that is greater than 1 is in the range values of x.

So the inequality is x>1.

Example 2

Consider the inequality -3x+7\geq4.

Solve the inequality.

Worked Solution

Create a strategy

Use the inverse operations to find the range values of x that will satisfy the inequality.

Apply the idea

\displaystyle -3x+7-7	\displaystyle \geq	\displaystyle 4-7	Subtract 7 from both sides
\displaystyle -3x	\displaystyle \geq	\displaystyle -3	Evaluate difference
\displaystyle \dfrac{-3x}{-3}	\displaystyle \leq	\displaystyle \dfrac{-3}{-3}	Divide both sides by -3, reverse inequality
\displaystyle x	\displaystyle \leq	\displaystyle 1	Evaluate

Now plot the solutions to the inequality -3x+7\geq4 on the number line below.

Worked Solution

Create a strategy

Plot the inequality solution from part (a)

Apply the idea

To graph the inequality x \leq 1, plot a solid dot on point 1 with a ray towards negative infinity.

Idea summary

To plot inequalities on the number line:

For \leq and \geq we use a filled in or closed dot to start the ray.
For < 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 satisfies the inequality and make sure your ray covers it.

Outcomes

MA5.2-8NA

solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques

3.04 Graphs of linear inequalities

Introduction

Ideas

Ineqeualities on the number line

Examples

Example 1

Example 2

Outcomes

MA5.2-8NA

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