Working with inequalities always gives us a nasty case of Pac-man fever. Wakka wakka wakka. Luckily, we know where to get our fix.
In some ways, an inequality is very similar to an equation. An inequation, if you will. Addition and subtraction work just like we would expect. We can add or subtract constants to both sides if we're solving for a variable:
x + 4 > 13
x + (4 – 4) > (13 – 4)
x > 9
Or we can add/subtract variables on both sides:
-6 ≥ -x – 1
(-6 + 6) + x ≥ (-x + x) + (6 – 1)
x ≥ 5
There are some important differences, though, when it comes to multiplication and division. For positive numbers, everything continues to be hunky-dory. If we have 3 ≤ 6 (and we do), then multiplying by 2 or dividing by 3 doesn't change a thing about the relationship.
6 ≤ 12
1 ≤ 2
Negative numbers, though, throw a wrench into our hunky-dory. Negative numbers, you should apologize to him. Try multiplying both sides 3 ≤ 6 by -2:
-6 ≤ -12
Or divide both sides of 3 ≤ 6 by -3:
-1 ≤ -2
See, that's completely messed up; just the total opposite of what is actually true. This happens every time we invite negative numbers over to multiply and divide. We've learned our lesson, though.
From now on, any time we multiply or divide by a negative number, we'll also switch the direction of our inequality signs.
Sample Problem
Solve -3x + 4 > x – 4.
It's like the song says: x to the left of us, constants to the right, here we are, stuck in the middle with a "greater than" sign. Okay, we may have taken some liberties with what the song says. Anyway, subtract x from both sides, then subtract 4 from both sides.
-4x > -8
Awoooga! Klaxons are going off here at Shmoop headquarters, warning us of mathematical danger. We're about to divide an inequality by a negative number. We proceed with caution.
x < 2
Whew, crisis averted. We switched the direction of the sign while doing the division dance. Failure to do so would have resulted in total failure, and obviously we don't want that.