Linear Inequalities

You are used to solving equations e.g. finding the value of x for which this is true
3x - 5 = 7
Treat it like a weighing scales, and do the same to both sides to solve the equation.
So

3x = 12 (+5 to both sides)
x = 4 (÷ 3 on both sides)

The same weighing scales analogy works for inequalities. If you have

3x - 5 ≤ 7
3x ≤ 12
x ≤ 4
In other words, the inequality is true for any number x as long as it is less than or equal to 4.
Try 4:
3(4) - 5 = 7 and 7 is ≤ 7
Try 2:
3(2) - 5 = 1 and 1 is ≤ 7
Try a number bigger than 4 e.g. 5 :
3(5) - 5 = 10 and 10 is not ≤ 7

When you solve an inequality you have to check which set of numbers the solution is in.
For example if you were told x ∈ N that means it is a natural number (whole number ≥ 0)
Then the solution set would be x = {0,1,2,3,4}
However, if it was an integer i.e. x ∈ Z, then the negative whole numbers could also be in the solution set.
Finally if you were told x ∈R, that means x can be any number (fraction, decimal or whole number).
In the last two cases you can't write out the solution set, so you will normally be asked to graph it on the number line.
To do this draw a numberline and indicate with dots and arrows which set of numbers are included.

NB Note that there is one important difference between solving equations and finding solution sets to inequalities.
If you have
-x = 3
then multiplying across by -1 gives you
x = -3
However, if you have
-x < 3
when you multiply accross by -1, you have to reverse the inequality
x > -3
The reason for this is simple. -1 < 2 but multiplying across by -1 gives 1 on the left and -2 on the right so the sign must turn around 1 > -2