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Equations and Inequalities

Equations and Inequalities

Checking Solutions to Equations

How Do We Check if a Value is a Solution to an Equation?

An equation works like this: (left-hand side expression) = (right-hand side expression). Therefore, an equation is only true if the left-hand side expression actually does equal the right-hand side expression. Take a look at your hands. Are they exactly the same?

It's totally normal if they aren't. Mostly, anyway. Um, maybe we'll come back to this line of reasoning later.

To check if a given value is a solution to an equation:

  1. Evaluate the left-hand side expression at the given value to get a number. 
  2. Evaluate the right-hand side expression at the given value to get a number.
  3. See if the numbers match.

Hey, it's matching! You do this with your socks every day. Sometimes not well, but at least the process is a vaguely familiar one.

If the numbers you get from evaluating the two expressions are the same, then the given value is a solution of the equation (makes the equation true). If the numbers don't match, the given value is not a solution of the equation (makes the equation false). Take those values that aren't solutions and dump them right in the trash, because we won't be needing them any longer. Actually, maybe rinse them out and put them in with the recycling instead. We're trying to be green.

Sample Problem

Is x = 5 a solution to the equation

The Not So Awesome Way (aka The Wrong Way)

If the first thing we do is write down

we are making a claim without having done the work to see if the claim is true. Oh snap.

The statement that the left-and right-hand sides are equal should come after evaluating the left-hand side, evaluating the right-hand side, and comparing the answers. If we were lawyers, we would call this our "due diligence." Luckily, this is Shmoop Algebra, and we're manfully resisting the urge to make horrible lawyer jokes right now.

The Super Awesome Way (aka The Right Way):

Evaluate the left-hand side for x = 5 to find that


Then evaluate the right-hand side for x = 5 to find that


Since 2 = 2, x = 5 is a solution to the equation. Bet knowing this will help you sleep better tonight.