Absolute Value Inequalities: Number Line Representation and Critical Point

Good news, test takers! You can pass the GMAT! Most of the questions are not that difficult. The difficulty in the test is in taking a mid-level question, (5x2 = 10) and rephrasing it in a more difficult way. (5 men walk into a bar. The bartender gives each man a beer for each of their hands. How many beers does the bartender distribute all together?) There is one exception to the rule, though: Absolute value inequalities.

It’s a little tough, but not impossible to figure out absolute value inequalities!

We’ll start with absolute value expression’s basic definitions, mathematically and conceptually.  Then we’ll use that basic understanding to solve equations and inequalities including absolute value expressions. Finally, we’ll deal with a concept-based application of the absolute value inequalities on the GMAT. Let’s go!

Absolute positive - Everybody knows that |9|=9 and|-9|=9.  What’s so special about those slanted lines? How and why does it neutralize a number?  Those bars are the measure of the distance of the number within the bars and zero. -9 and 9 are both 9 units away from 0. Picture it!


Let’s look at it those two ways

1) The number within the bars will definitely turn positive and

2) The distance from zero to the number is our basis for digging deeper into absolute value.

This concept becomes difficult when a variable, as opposed to a number, is placed within those bars, |u|. Shouldn’t |u| be equal to u? There’s no sign in front of the u, so absolute values says this u should be positive…but this u could actually be negative!

When dealing with absolute values of variables, know these constants

1) the expression (number or variable) may have a positive sign [+]

2) the expression (number or valuable) may have a negative [-] sign.

The Bottom Line: Numbers or variables with the absolute value bars can be negative or positive.

Solving absolute values

Step one: Remove the bars…but before you do that, consider the possibility of a negative or positive scenario. |u| + 6 =10

Scenario 1: If u is positive, simply drop the absolute value bars and rewrite the equation as u+6=10, u =4

Remember that if u is greater than 0, then u=4

u>0, then u=4

Scenario 2: If u is negative, removing the absolute value bars negates [-] the expression. The equation is then written (-u)+8=12, u=-4

u<0, u=-4

Now remember the “5 men walking into a bar” comparison. They want to complicate a simple mid-level equation here, so be very careful. |u+1| =8

SOLVE

Scenario 1:

Remember that

u+1 is positive, u+1=8,  so u=7

Don’t forget that if

u>-1, then u=7

Scenario 2:

(u+1) is negative (this time)

Then –(u+1)=8 and u=-9.

Therefore, u<-1, and u=-9

Did you see in those two examples that there are two solutions to the GMAT inequalities with absolute value equations?

This is because a variable absolute expression depends on the value (positive or negative.) Each variable has a breaking point, known as the critical point. The critical point is the zeroing point of the values in the absolute value bars.

For |u|, the critical|zeroing point is 0.

All u’s to the right of the 0 on the number line changes the variable:
|u| to u

All the u’s to the left of the number line changes the variable: |u| to (-u)



For the expression, |u-2|, 2 is the critical point.

For the expression, |2u+3|, -3, 2 is the critical point.

The critical point is important because it tells the numerical range that makes each equation true. GMAT inequalities with absolute values are one of those topics that teachers use to try and trick unsuspecting students. With a little bit of direction and awareness, you’ll be fine. You only need a little bit of knowledge and the fundamentals. You’ll solve these no-so-tricky problems with ease.