We here at Shmoop are big fans of equality. It makes life, and math, much easier to deal with. Sometimes, though, we can't escape the inequalities of life.
The word "inequality" basically means that a number is "inequal" to, or unequal to, or not equal to, or not the same as, another number. In other words, it can be "greater than" or "less than" that number: > or <.
But not (>_<). That's an emoticon.
Some inequalities allow just a little bit of equality: ≥ and ≤. Those are "greater than or equal to" and "less than or equal to." We'd prefer full equality (and a milkshake, and a million dollars, and world peace, and and and), but baby steps.
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.
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.
Our old man's cooking is really hit or miss. Sometimes it's fantastic, and sometimes it's completely horrible. It's never in between, though:
Papa Shmoop's cooking > our favorite restaurant
Papa Shmoop's cooking < dog food
This is a compound inequality, which has more than one inequality holding it down. This time, a solution to the whole thing can satisfy either the first inequality or the second. We're not picky about which.
Here's another example, now with more of those number thingies:
x + 1 < -2 or -2x < 2
To solve a compound inequality joined by "or," we solve each mini-inequality separately.
x + 1 < -2
x < -3
For the second inequality:
-2x < 2
x > -1
Don't forget to switch the signs around when dividing by a negative number. We just went over this.
With the separate inequalities solved, we join them back together at the hip.
x < -3 or x > -1
If x is less than -3, we're golden. If x is greater than -1, that's cool too. However, if x falls into the shark-filled pit in the middle, there is no hope. We suggest averting your eyes if that happens.
Instead, look at the number line for this inequality. It too shows that the full inequality is true when one or the other inequality is true.
We can also have inequalities joined together by the word "and." This time, both inequalities have to be true for a number to be a solution.
x > -2 and x < 5
Unlike "or" inequalities, we can smash an "and" inequality into one giant, wordless math expression, and that's usually how we encounter them. The variable should sit in the middle of the inequalities. We want the smallest number on the left side of the inequality, and the largest on the right side.
-2 < x < 5
We still read this as "x is greater than -2 AND less than 5." Despite appearances, nothing has actually changed. Also, there is no spoon. We only eat with sporks these days.
We can double-check that we've arranged everything right by looking at which way the hungry inequality mouths point.
Since x > -2 has the "mouth" pointing away from -2 and towards x, it's saying x is bigger. For x < 5, though, Pac-man fever has gripped 5, leaving x in the dust. That's exactly what we see in our combined expression as well.
There are a lot of numbers greater than -2, and a whole mess of them less than 5. However, the only numbers that are both at the same time are nestled right here. Best get cozy with them, if we want to be friends with the compound inequality's solution.
All of the rules for solving inequalities stay the same when we have a compound inequality joined by "and." We just need to expand on them a bit. For instance, we can add 3 to every side of the inequality, and all of the inequalities will remain equally unequal.
Likewise, when we multiply by a negative number, we switch everything around: positive and negative signs, greater than and less than signs, even Zodiac signs. We're a Serpentarius now, but for some reason our astrology reading is still the same.
20 > x > -16
We want to reorder the joined inequality to go from lowest to highest. Shuffling everything around can be confusing, unless we remember to focus on where the mouths are pointing.
-16 < x < 20
That's the ticket.
Simplify x ≤ 3 and x > 7.
This seems like a simple problem. Maybe too simple. We smell a trap. Or is it Grandma's tuna casserole? Whatever, we'll combine these inequalities together, from smallest to largest.
3 ≥ x > 7
Any solution has to be smaller than 3, like 0, and greater than 7, like 10. Houston, we have a problem. The compound inequality wants the impossible from us. Like two siblings fighting over what to watch on TV, there can be no solution here.
In general, if we put everything in order, smallest to largest, then an "and" compound inequality should have all of its "mouths" pointing to the right. Think of it like a buffet line with all the best stuff at the end. Those inequalities can't help themselves from staring and salivating. Uh, gross, wipe up after yourselves, please.