Inequalities

Lesson

Previously, we were introduced to the four main types of inequalities. Here are some examples of these types:

$x<2$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" |

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 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. For example, if we know that $x=6$`x`=6, we can plot the value of $x$`x` as follows:

Similarly, if we know that $x=\frac{19}{5}$`x`=195, we can plot the value of $x$`x` as follows:

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

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:

So far so good. But what about fractions like $x=\frac{1}{2}$`x`=12, or irrational numbers like $x=\sqrt{2}$`x`=√2? 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 rational and irrational 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 (a directed line) to represent all of these points, since **all of them** are included in the inequality.

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:

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 $x\ge4$`x`≥4 and $x>4$`x`>4 are plotted below:

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

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!

Plot the inequality $x<0$`x`<0 on the number line below. Make sure to use the correct type of endpoint.

Plot the inequality $x<2$`x`<2 on the number line below. Make sure to use the correct type of endpoint.

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