Plot further inequalities

Previously, we were introduced to inequalities. There are four main types of inequalities. Here are examples of each type.

$x<2$x<2	"$x$x is less than $2$2"
$x>-5$x>−5	"$x$x is greater than $-5$−5"
$x\le-4$x≤−4	"$x$x is less than or equal to $-4$−4"
$x\ge17$x≥17	"$x$x is greater than or equal to $17$17"

We are now going to see how we can represent inequalities on the number line.

The Number Line

Remember that all the real numbers can be represented on an infinite line called the number line, stretching all the way from negative infinity to positive infinity. 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 labelled). 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. Say we were given the equation $x=6$x=6. We could plot $x$x like so.

Here are some other examples of real numbers we could plot on the number line.

$x=\frac{19}{5}$x=195​
$x=-\sqrt{2}$x=−√2
$x=2\pi$x=2π

Inequalities on the Number Line

Now, what if we wanted to plot an inequality, such as $x\le4$x≤4?

When you say "$x$x less than or equal to $4$4", you're not just talking about one number. You're talking about a whole set of numbers, such as $4$4, $2$2, $3$3, $0$0, $-1$−1, $-1000$−1000, $\frac{1}{2}$12​. All of these numbers are less than or equal to $4$4.

So should we plot the inequality $x\le4$x≤4 like this?

The problem with this is that it's not just these numbers that are less than or equal to $4$4. It's the countless collection of numbers in between, along with every number down towards negative infinity. One endless line of numbers to the left of $4$4. We represent this as a ray from $4$4 pointing all the way left across the entire number line.

$x\le4$x≤4

Now, what if instead we wanted to plot the inequality $x<4$x<4. The only difference now is that $x$x cannot be equal to $4$4. To represent this, we plot the ray in the same way, but since $4$4 is no longer included, we have a hole where $4$4 is supposed to be.

$x<4$x<4

For $x\ge4$x≥4 or $x>4$x>4, we just flip the direction of the ray!

$x\ge4$x≥4
$x>4$x>4

Here are some other examples of inequalities plotted on a number line.

$x\ge-\sqrt{3}$x≥−√3
$x<\pi$x<π
$x>-\frac{63}{67}$x>−6367​
$x\le-20.7$x≤−20.7

Worked Examples

Question 1

Question 2

Question 3