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.)