Why can we solve an inequality like an equation when solving, and could you provide some examples?

C

Cambridge101

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I would like to know why it holds that doing the same thing to both sides of an inequality works. In my head its clear, I would just like some condensed clarity from someone else, how would you explain it?

For example, 10>5

if I just take 1 from 10, then you get 9>5 this still holds despite only taking 1 from 10. In more complex examples, does it have anything to do with a common difference or common ratio. 5/10 = 1/2 is not the same relationship as 5/9?
 
In an equation, when you do the same thing to two equal numbers, the results are still equal. if 2x = 6, then when we divide both sides by 2, the results are still equal, so x = 3.

In an inequality, most things you can do to both sides retain the same relationship. Imagine a balance scale; if you have 10 grams on one side and 5 grams on the other, then taking away 1 gram from each side will not change the fact that the left side is heavier. Then same is true if we add to both sides, or double both sides, and so on.

The cases where that isn't true are when you multiply or divide by a negative number (which you can't picture with a scale!). That reverses everything. I like to picture this using a number line. Adding or subtracting the same thing to two numbers just slides them along, keeping the same order; multiplying by a positive number just stretches things, again retaining order; but multiplying by a negative number flips everything around, and it goes backward.
 
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