Inequalities

Ontario 10 Academic (MPM2D)

One step inequalities on a number line

Lesson

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.

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.

Consider the inequality $2x>-4$2`x`>−4.

Solve the inequality.

Now plot the solutions to the inequality $2x>-4$2

`x`>−4 on the number line below.

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.