3. Algebra and Equations

Lesson

Previously, we were introduced to the four inequality symbols, and learnt how to solve simple inequalities. Here are some examples:

$x<2$x<2 |
"$x$x is less than $2$2" |

$x>-5$x>−5 |
"$x$x is greater than $-5$−5" |

$2x\le-4$2x≤−4 |
"$2$2 groups of $x$x is less than or equal to $-4$−4" |

$x-3\ge17$x−3≥17 |
"$3$3 less than $x$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 as learnt in Grade 7.

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 smaller numbers and numbers further to the right are larger 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 with a solid dot:

Similarly, if we know that $x=\frac{19}{5}$`x`=195, we can plot the value of $x=3\frac{4}{5}$`x`=345 as follows:

Now, what if we wanted to plot an **inequality**, such as $x\le4$`x`≤4? We can review this from Grade 7.

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?

To show all of the values less than or equal to $4$4, 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 examples of inequalities plotted on a number line that have negative values:

Remember!

- For $\le$≤ and $\ge$≥ we use a filled in or closed dot to start the ray, to show the starting point is included
- For $<$< and $>$> we use a hollow or open dot to start the ray, to show the starting point is
**not**included - To check your ray is going the right way, choose a value which satisfies the inequality and make sure your ray covers it

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

Plot the inequality $x\le1$`x`≤1 on the number line below.

Consider the inequality $-3x+7\ge4$−3`x`+7≥4.

Solve the inequality.

Now plot the solutions to the inequality $-3x+7\ge4$−3

`x`+7≥4 on the number line below.

Solve inequalities that involve integers, and verify and graph the solutions.