As we have previously seen, we can plot inequalities by using number lines.

For example, a plot of the inequality $x\le4$x≤4 looks like this:

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 $x\le4$x≤4 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:

Multiplying or dividing both sides by a negative number will reverse the inequality symbol.

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 1

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

Solve the inequality.

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

Question 2

Consider the inequality $2x>-4$2x>−4.

Solve the inequality.

Now plot the solutions to the inequality $2x>-4$2x>−4 on the number line below.

Question 3

Consider the inequality $\frac{x}{-7}<2$x−7<2.

Solve the inequality.

Now plot the solutions to the inequality $\frac{x}{-7}<2$x−7<2 on the number line below.