# 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.

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

x>=5/6
x<=5/6