Algebraic Inequality

Normally, I'm in favor of equality. But not today. Today we're talking algebra.

Don't let the "algebra" word trick you: inequalities are easy. Picture one of those old-style balances, with a plate on each side. You put something on the left and something on the right, and whichever side is heavier goes down.

Let's say you have 8 pounds on the left, and y pounds on the right (hey, I said it was algebra), and the scale balances. OK, this one's easy: y = 8.

Let's make it a little harder. Now we're going to have 14 pounds on the left, and y pounds on the right. We look at the scale and see that the right side is heavier. That's the one with y, so we can say y is more than 14:  y > 14.

OK, that's pretty easy. What will happen if we add two pounds to both sides? Umm...nothing, right? The right side was heavier before we added the two pounds to both sides, and it's still heavier now: y + 2 > 16.

What if we go back to y > 14 and multiply both sides by two? (We put 14 more pounds on the left and y more on the right?) We know what's going to happen there: nothing again. The right side was heavier before, and it still is: 2y > 28.

What if we multiply by negative 1 instead? This is trickier to picture. Before the left side was being pulled down by 14 pounds, and the right side was being pulled down by more than that (y pounds, whatever that is). Now we're going to pull up on the left side with 14 pounds, and pull up on the right side with y pounds. We're pulling the right side up harder, so the right side will be up and the left side will be down. (Do you see that?)

OK, so what have we learned? If you have an inequality like

y > 14

we can add (or subtract) any number from both sides without changing the greater than (or less than) sign. We can multiply by a positive number without changing the sign. But if we multiply by a negative number, the sign will flip:

-y < -14

Let’s fill in a number that works in the original equation and make sure it still makes sense:

                y > 14

Let’s pick a number for y. It doesn’t mean it’s the one correct number, because this isn’t an equality. It’s an inequality, and we’re just picking one of the things y might be. Maybe y is 20: that’s more than 14. And we said 

                -y < -14

Does that still work for 20?
                -20 < -14 (Yes, it does.)

So let’s look a few problems we can now solve.

x + 38 > 50

Well, we know that we can add or subtract anything from both sides, and we really kind of want the x by itself. So we can subtract 38 from both sides.

                x + 38 - 38 > 50 - 38

                x > 12

How about 3y - 18 < 6 ?

Hm. It would be nice to just have 3y on one side, so let’s add 18 to both sides:

                3y < 6 + 18    (note that on the left I got rid of the “-18” on the left side by adding 18)

                3y < 24

Then we can divide both sides by 3. That’s the same as multiplying by 1/3, which is positive, so the sign doesn’t change:

                y < 8

Are we sure we’re right? Our answer here tells us that if we put 7, it will satisfy the original equation:

                3y  - 18 < 6

                3(7) - 18 < 6

                21 - 18 < 6

                3 < 6 (Well, yeah)

Our answer tells us that if we put 9, it won’t satisfy the original equation:

                3y - 18 < 6 (this should be false)

                3(9) - 18 < 6

                27 - 18 < 6

                9 < 6 (It’s not, just as we thought.)