# Why do you not change the inequality sign when you are adding or subtracting?

Jul 24, 2018

Because to do so would be algebraically incorrect. See below.

#### Explanation:

Consider the simplest of inequalities: $a < b$ $\left\{a , b\right\} \in \mathbb{R}$

Now consider adding or subtracting a real number, $x \in \mathbb{R}$ to the LHS. $\to a \pm x$

The only way to restore the inequality is to add or subtract $x$ on the RHS.

Thus: $a + x < b + x \mathmr{and} a - x < b - x$ both follow from the original inequality. To reverse the inequality would simply be incorrect.

So when must we reverse the inequality?

Consider where we multiply (or divide) both sides of the inequality by $x < 0$ (i.e. any negative real number)

As an example I will use $x = - 1$

Then, if $a < b \implies a \times \left(- 1\right) > b \times \left(- 1\right)$

So, in order to maintain the inequality after multiplying or dividing through by a negative number we must reverse the inequality.

Hope this helps. It's not as complicated as it seems!