Manipulating inequalities

[yo4561]

I am confused on the rule of subtracting and adding any value to both sides of an inequality in that the inequality remains valid. Why can we randomly do this? Can you do this for multiplication and division too (e.g., multiply both sides of the inequality by the same number and have it remains valid)?

Thank you for your time and help

[TiffanyB]

I am confused on the rule of subtracting and adding any value to both sides of an inequality in that the inequality remains valid. Why can we randomly do this?

Hello yo4561,

I think it's helpful to think of this as a math rule. In other words, it's not random that we can do this, it's simply a rule of mathematics.

You can add or subtract to both sides of an inequality without changing that inequality because inequalities express relationships. If x > y, for example, then x could be 2 and y could be 1. Adding any number to both of these numbers will not change their relationship. Add 5 to both for example; then you have 7 > 6. That's still true! Subtracting is similar.

Can you do this for multiplication and division too (e.g., multiply both sides of the inequality by the same number and have it remains valid)?

Thank you for your time and help

You can multiple and divide on both sides of an inequality as well; however, the rule is a little more complicated.

If you know that you're multiplying or dividing by a positive number, do the same as you would when adding and subtracting.

However, if you know that you're multiplying or dividing by a negative number, you'll need to switch the inequality sign when you do this. Test some real numbers to understand this better. Try x > y and set x = 2 and y = 1. If you divide by -1 on both sides, then you'd have -2 > -1. That's not true, though, because numbers are considered lesser as they become more negative. So when you divide by -1, you'd switch the sign. Then you'd get -2 < -1. This is true.

If you're multiplying or dividing by a variable and you're not sure whether it is positive or negative, you'd need to test two cases: one in which the variable is positive and one in which the variable is negative. This can get quite complicated, so there is often a different and better path forward than doing this.