Eduki

The process of solving inequality and equation isn't very different from each other. The additive and multiplicative laws of equality are both applicable in the case of inequality as well. The difference is only when using the multiplicative law of equality with a negative number. For example, we have an inequality that states that a number deducted from 7 is less than 8. If the assumed number is “a”, then 7-a<8

Deducting 7 from both sides, we get

7-a-7<8-7

-a<1

Multiplying both sides by -1

a<-1

Now, let's check the solution found. If we put any number that is less than -1 in the given inequality such as a= -2 then,

7-(-2)=9 which is greater than and not less than 8. 

This shows that the solution is erroneous. The actual solution is the opposite of what seems.

To understand that let's take a numerical inequality.

 5>3

If both sides are multiplied by -1

-5 is less than -3 instead of -5 is more than -3. 

Therefore, -5<-3

To avoid confusion, the inequalities are solved as an equation by replacing the inequality sign with an equal sign and then later checking the solution with testing points.

Open and darkened circles

There are cases in number lines where the circle shown above is darkened instead of not getting filled at all. One example is 

In such cases, the solution is x≥-4 instead of x>-4 and the difference is that the solution includes -4 as well along with all points to the right of -4 in x≥-4 while in x>-4 there are all points to the right of -4 but not -4 itself. 

The process of testing with a test number becomes very important in negative inequalities - inequalities that contain negative coefficients of the variables.