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:
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 for $x$x.
Now plot the solutions to the inequality $3+x<2$3+x<2 on the number line below. Make sure to use the correct type of endpoint.
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.