Solving Inequalities
Sometimes we need to solve inequalities like these:
Symbol 
Words 
Example 



> 
greater than 
x+3 > 2 
< 
less than 
7x < 28 
≥ 
greater than or equal to 
5 ≥ x1 
≤ 
less than or equal to 
2y+1 ≤ 7 



Solving
Our aim is to get "x" (or whatever variable is there) to be on its own.
Something like: 

x < 5 
or: 

y ≥ 11 
Adding or Subtracting a Value
We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra), like this:
Solve: x + 3 < 7
If we subtract 3 from both sides, we get:
x + 3  3 < 7  3
x < 4
And that is our solution: x < 4
In that example, x can be any value less than 4. This is how it looks on the number line:

x can go up to (but not equal to) 4,
so that x+3 goes up to (but not equal to) 7 
And that works well for adding and subtracting, because if you add (or subtract) the same amount from both sides, it does not affect the inequality
Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.
What if I solve it, but "x" is on the wrong side?
No matter, just swap sides, but reverse the sign so it still "points at" the correct value!
Example: 12 < x + 5
If we subtract 5 from both sides, we get:
12  5 < x + 5  5
7 < x
Let us flip sides (and the inequality sign!):
x > 7
Do you see how the inequality sign still "points at" the smaller value (7) ?
And that is our solution: x > 7
Multiplying or Dividing by a Value
Another thing we can do is to multiply or divide both sides (just as in Algebra  Multiplying).
But we need to be a bit more careful (as you will see).
Positive Values
Everything is fine if you want to multiply or divide by a positive number:
Solve: 3y < 15
If we divide both sides by 3 we get:
3y/3 < 15/3
y < 5
And that is our solution: y < 5
Negative Values

But if you multiply or divide by a negative number you have to reverse the inequality. 
Why?
Well, just look at the number line!
For example, from 3 to 7 is an increase, but from 3 to 7 is a decrease.

7 < 3 
7 > 3 
See how the inequality sign reverses (from < to >) ? 
Let us try an example:
Solve: 2y <8
Let us divide both sides by 2 ... and reverse the inequality!
2y/2 > 8/2
y > 4
And that is the correct solution: y > 4
So, just remember:
When multiplying or diviing by a negative number, reverse the inequality
Multiplying or Dividing by Variables
Here is another (tricky!) example:
Solve: bx < 3b
It seems easy just to divide both sides by b, which would give us:
x < 3
... but wait ... if b is negative we need to reverse the inequality like this:
x > 3
But we don't know if b is positive or negative, so we can't answer this one!
To help you understand, imagine replacing b with 1 or 1 in that example:
 if b is 1, then the answer is simply x < 3
 but if b is 1, then you would be solving x < 3, and the answer would be x > 3
So:
Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).
A More Complicated Example
Solve: (x3)/2 < 5
First, let us clear out the "/2" by multiplying both sides by 2:
(x3)/2 ×2 < 5 ×2
(x3) < 10
Now add 3 to both sides:
x3 + 3 < 10 + 3
x < 7
And that is our solution: x < 7
Conclusion
Many simple inequalities can be solved using the techniques of adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
