Inequalities

We are now looking at solving inequalities. These are very similar to our standard equations, but they have < or > instead of = signs.
The < or the > stand for greater/less than. This depends on the way the sign is facing. I like to think of the < or the > as a mouth, with the larger open end the opening of the mouth. The mouth always wants to eat the larger meal, so it always goes for the larger part of our inequality. So if we have 3x +2 > 2x - 1, the "mouth" is trying to eat the "3x + 2" so this must be the larger part. So we can rewrite this equation as in words as "3x + 2 is greater (larger) than 2x - 1".
Now there are slight modifications that can happen to these. We can write them as ≥ or ≤. The change is the line underneath. This changes it from being "greater/less that" to "greater/less than or equal to". So for example; x ≥ 4 means that x could take any value greater than OR equal to 4. 


We can USUALLY solve these in the same way as we would solve standard equations. So with the following example:

Solve 3x + 4 > 7

We want x on one side of our > and everything else on the other side. So, we will first subtract 4 from both sides of our equation. We get;

3x + 4 - 4 > 7 - 4
3x > 3

Now we divide both sides by 3. We get;

3x/3 > 3/3
x > 1

So for the equation 3x + 4 > 7 to be true, x must be greater than 1 (x>1)


Now you would see that I said we only USUALLY solve the equations in the same way. The exception to this is when we divide or multiply by a negative number. We have to do something a little different with these. If we do ever have to multiply or divide by a negative, we need to switch the way the > or < sign is facing. So, < becomes >, or visa versa.

For example;

Solve 7 - 4x > 15

First we will get rid of the 7, by subtracting 7 from both sides. We get;

7 - 4x - 7 > 15 - 7
-4x > 8

Now, we will divide by -4, and because this is a negative we must switch the direction of the > (so it will become <). We get;

-4x/-4 < 8/-4
x < -2

So we have now solved this, giving us a solution of x being any number less than -2 (or x < -2)


Now, we also need to know how to plot these on a number line. To do these, I have found that we can simply follow 3 steps.

1. Draw an open circle at the number on our number line, i.e. if we had x>1, we would draw an open circle at the point 1 on our number line. 
2. Colour this circle in if it is a ≥ or a ≤ 
3. Draw a line with an arrow in the correct direction. 

Just as a reminder, this is what a number line looks like:

So for example, plot x > 2 on the number line; 

First we draw our open circle at the point 2 on our number line like so;

Next, we look to see if our sign is a ≥ or a ≤. In this case it isn't, so we don't have to colour our circle in. 

Now, we want to draw a line with an arrow in the correct direction. In this case, the line needs to represent x, which is any number larger than 2 (x>2), so our line and arrow has to cover any number greater than 2. So it will look like; 

 Done!

Another example could be x ≤ -1

First, we draw a circle on our number line at -1. 

Now since this one does have a ≥ or a ≤, we have to colour in the circle. This gives us; 

Finally, we draw the line and arrow in the direction required. In this case, the equation is telling us that x is any number less than or equal to -1, so we have to draw our line and arrow to represent this. We then get; 

A great website for more information and examples on this is this one: 

http://www.mathsisfun.com/algebra/inequality-solving.html

I have uploaded a worksheet on this topic into the "Work Sheets" tab. You should now be able to work through these questions. They need to be completed by Wednesday next week. 

As always, if you have any questions about this, feel free to comment on this, or email or chat to me in person.