Last week we talked about “he or she” questions in the writing section, a rare example of something can “sounds” wrong but is actually grammatically correct.

Today, I will once again veer off in a completely different direction and talk about one of math’s more annoying and frustrating concepts: inequalities.

Inequalities have always been a thorn in my side, and their presence can muddle even the most straightforward of questions.

First of all, a very brief review: inequality symbols show the relationships between numbers, variables, or a combination of the two.  For example, when I say x > 5, I mean that x is larger than 5 (just remember, the alligator eats the bigger one).  That means x could be 6, 7, 100, 5.00001, or anything else that is larger than 5.  Conversely, if I say y < 7, that means that y is smaller than 7 (2, 4, -100, 6.9999, or anything else less than 7).

Easy enough, right?

Unfortunately, not really.

A big problem with inequalities is they throw people off and make otherwise easy questions seem more difficult.  Imagine the following problem.

4(2x-5) > 6x + 2

Most of my students would have little trouble solving this equation if there were an equal sign in the middle.  Therefore, the biggest piece of advice I give students (and something that I do myself) is to imagine the inequality symbol is an equal sign and just solve it normally.

Ok, so normally with equations like the one above, we want to get x by itself, so lets just try to do that!

                 8x – 20 > 6x + 2 (Distributive property)

                   2x – 20 > 2 (Subtract 6x from both sides)

2x > 22 (Add 20 to both sides)

x > 11 (Divide both sides by 2)

Really not so terrible.  x is greater than 11.  Approaching these problems like a normal equation is a small, subtle strategy that might not necessarily have any objective “benefit” but definitely helps me psychologically when solving for variables.

However, there is one big difference between solving inequalities and solving equations: If you multiple or divide both sides by a negative number, you must flip the sign!

To show you how this works, let me solve the above question again, but I will solve it a bit differently so that we get some negatives.

                  8x – 20 > 6x + 2 (Distributive property)

                   – 20 > -2x + 2 (Subtract 8x from both sides)

      -22 > -2x (Subtract 2 from both sides)

                                        11 < x (Divide both sides by -2 AND flip the sign around)

You’ll notice that once again we get x is greater than 11 (which we should get again, since we’re solving the exact same equation).  This is the one big difference between solving inequalities and solving equations.

As you can imagine, students forget to do this (or do it wrong) quite often when solving inequalities, so I generally encourage students to avoid multiplying or dividing by negatives if possible.

Something people get mixed up about, however, comes with questions like the one below.

4x > -8

Ok, so to get x by itself, we’re going to divide both sides by 4.  Now the question is, do we need to flip the sign here?

NO!!!

Even though the number that gets divided is negative, the sign stays the same because we are still dividing BY a positive number).

So the answer to the above question is x > -2.  Notice we keep the sign in the original direction, since we didn’t divide BY a negative.

To conclude, inequalities cause many students to make many mistakes, but these mistakes are very avoidable if you just solve the equations normally and flip the sign if you multiply of divide by a negative number.

There was an actual SAT question with inequalities I wanted to show, but it seems that will need a blog post of its own.

Till next time!

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