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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 ≥ x-1
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

warning! 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: (x-3)/2 < -5

First, let us clear out the "/2" by multiplying both sides by 2:

(x-3)/2 ×2 < -5 ×2  

(x-3) < -10

Now add 3 to both sides:

x-3 + 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.