Why do we change the sign of inequality in negatives?

Answer 1

If guess you mean when doing algebra you have something like:

-x > 5

and you change it to

x < -5

when with an equals sign it makes no difference.

What you need to consider is what you are actually doing when you change the sign of an expression:

You are actually adding the same amount to both sides of the inequality:

-x > 5

→ -x + x > 5 + x

→ 0 > 5 + x

→ 0 > x + 5

Or subtracting the same amount from both sides:

→ 0 - 5 > x + 5 - 5

→ -5 > x

This is read (left to right) as: "-5 is greater than x"

Now if -5 is greater than x, then x must be less than -5, which would be written as: x < -5

The overall effect is to make it look like both sides have been multiplied by -1.

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It is only our convention for reading left to right that makes the '>' symbol to be known as "greater than", and the '>' symbol to be known as "less than":

• x > y is read as "x is greater than y"
• x < y is read as "y is less than x"

It is the end of the symbol which is wider which gives the greater expression.

They can, however, be read right to left, but KEEPING the expression at the wider end as the greater one:

• x > y can also be read as "y is less than x"
• x < y can also be read as "y is greater than x"

As they mean the same thing.