Equations and Inequalities
Introduction to :
Aw...what did they ever do to you?
When lots of fractions are involved, there is another way to make an equation look simpler before solving it: get rid of the fractions. Sweep them away, pack them in garbage bags and dump then into the bay. But not really, because that's littering. To get rid of the fractions, we pick a useful number and multiply both sides of the equation by that number. The number is useful if multiplying eliminates all fractions.
Plus, if the number does a good enough job cleaning up the fractions, maybe we'll see how it does with our bedroom.
Solve the equation .
Way 1. Subtract 2/3 from each side, so that , and add the fractions.
Way 2. Find the LCD of the fractions—in this case, 6. Multiply the left-hand side of the equation by 6 and the right-hand side of the equation by 6 to find that
Oy. So many fractions and parentheses, we better simplify this sucker. Fortunately, it simplifies to
4 + 6x = 1
Much better. Notice that there are no longer fractions in the equation, not to mention that the parentheses are gone as well, which is a nice bonus. Subtract 4 from both sides, and we see that
6x = - 3
so is the solution to the equation. We wound up with a fraction anyway, but it sure was nice being without them at least for a little while.
Make sure you understand that getting rid of fractions isn't the same thing as "simplification." When we "simplify," we rewrite the expressions on each side of the = sign to be tidier, but we don't change the value of either expression. When we eliminate fractions, we are multiplying both sides of the expression by the same number and therefore changing the values of both expressions—but in such a way that the scale is still balanced. Each side is much, much heavier. In fact, we should probably put the whole thing on a sturdier table.