# Inequalities with Addition and Subtraction

## Key Questions

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!

inequality with or without fractions are solved using the negative signs to reverse the direction of the inequalities.

#### Explanation:

Fractions do not change the way inequalities are solved. Treat the fractions the same way that fractions would be solved in an equation.

Fraction do not change the way inequalities are solved remember to reverse the direction of the inequality whenever the inequality is multiplied or divided by a negative sign.

Negative means to do the opposite a $- \times - = +$
a $- \times + = -$
a $- \times < = >$