Inequalities and dividing by a variable

Variable – is a symbol for a number we don’t know yet. `x` is the variable in the term `4x`.

If your equation is

`3x\ 4y<=10y` 

You cannot divide by `y`  to get `x` on its own because:

How can you divide if you don't know what `y` is?

Is it +ve value or -ve value?

Depending on whether `y`  is +ve or -ve will determine whether the inequality is flipped or not.

Example

`3x\ 4y<=10y` 

1.  If `y=-2` then:

`3x\ 4y<=10y`

`3x\ 4(-2)\<=10(-2)`

`3x\ (-8)\<=-20`

`-24x<=-20` 

Divide both sides by -24 and flip the inequality

`(-24x)/-24>=(-20)/-24`

`x>=(-20)/-24`

`x>=10/12`

`x>=5/6`

2.  If `y=+2` then:

`3x\ 4y<=10y`

`3x\ (4xx2)<=10(2)`

`3x\ 8<=20`

`24x<=20`

Divide both sides by 24 but DON’T flip the inequality

`(24x)/24<=20/24`

`x<=10/12`

`x<=5/6`

The two answers give:

`x>=5/6`

Or

`x<=5/6`

You can’t say which it is.