Solving Inequalities by Multiplying or Dividing (12-6)

Solving Inequalities by Multiplying or Dividing

Examples

Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying).
But we need to be a bit more careful (as you will see).

Positive Values

Everything is fine if we 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

When we multiply or divide by a negative number we must 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 < -8

-2y/-2 > -8/-2

y > 4

And that is the correct solution: y > 4

(Note that I reversed the inequality on the same line I divided by the negative number.)
So, just remember:

When multiplying or dividing 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 we 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 Bigger Example

Solve: (x-3)/2 < -5

First, let us clear out the "/2" by multiplying both sides by 2.
Because we are multiplying by a positive number, the inequalities will not change.

(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

Two Inequalities At Once!

How do we solve something with two inequalities at once?

Solve:

-2 < (6-2x)/3 < 4

First, let us clear out the "/3" by multiplying each part by 3:
Because we are multiplying by a positive number, the inequalities will not change.

-6 < 6-2x < 12

Now subtract 6 from each part:

-12 < -2x < 6

Now multiply each part by -(1/2).
Because we are multiplying by a negative number, the inequalities change direction.

6 > x > -3

And that is the solution!
But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):

-3 < x < 6

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

Video Help

https://www.youtube.com/watch?v=PNXozoJWsWc

https://www.youtube.com/watch?v=3ewLnYIQ6QE

https://www.youtube.com/watch?v=r3zdr8DyeL8