Intuition: Solving Equations vs. Solving Inequalities

One of the most fundamental skills in math is the ability to solve equations. The rules to the game are simple:

You are able to perform all of the four basic operations ($+$, $-$, $\times$ $\div$) on either positive or negative, real or non-real values (just remember, subtracting a negative value is the same as adding that value, and dividing/multiplying two negative value creates to a positive value! #doublenegative).

and

When you finish solving the equation, there must exist at least one real or non-real solution (the amount of solutions you might find depends on the degree of the polynomial).

Let's do some examples

Example 1.) Solve $x^2 + 11x + 18 = 0$

Good, let's try something harder

Example 2.) Solve the equation $\frac{5}{2-x} + \frac{x-5}{x+2} + \frac{3x+8}{x^2-4} = 0$

Simple, right? Well the name of the game has changed to solving inequalities! This can be considerably tough thing to do because our two rules have changed a bit...

You may perform all of the four basic operations on any positive value. HOWEVER, if you multiply or divide both sides of an inequality by a negative value, you have to reverse the direction of the inequality sign!

and

When you finish solving the equation, the inequality must hold true for all solutions that are within the boundaries of that inequality. It must also be that all values that are not within the scope of our inequality, must not be solutions (This is relieved if the inequality serves to be a condition for solving an equation).

Let's finish up with another set of examples:

Example 1.) The velocity of an object fired directly upward is given by $V = 80 – 32t$, where $t$ is in seconds.

Cool, one more!

Example 2.) You want to invest $30,000. Part of this will be invested in a stable 5%-simple-interest rate account. The remainder will be "invested" in your father's business, and he says that he'll pay you back with 7% interest. Your father knows that you're making these investments in order to pay your child's college tuition with the interest income.

So the moral of the story is: You can have all the fun you want, but follow the rules and be careful!

P.S. For those of you who have mastered these skills and wish to give yourselves a bit of a challenge, then try the exercises below!