How to Solve Absolute Value Inequalities in GMAT

The GMAT focuses on measuring the candidate’s ability to solve quantitative problems and interpret data. The concept of absolute value inequalities have been known to be the most challenging. With the GMAT being a tightly timed examination, it is crucial that candidates dissect the questions quickly and come up with the correct answers. Let us look at the secrets of mastering absolute value inequalities.

An inequality is a comparison between two values or expression where either values or expressions are not equal to one another. Most of the algebraic manipulations that are used in mathematical expressions can be applied to inequalities; however, there are several exceptions listed below that need to be applied. These points may seem trivial; however, the consequences can be severe, and can mislead candidates to the wrong answers.

1) When an inequality is being multiplied by a negative value, the direction of the sign must be switched.

2) Unlike expressions, inequalities cannot be squared. This means that a > y will not equal to a^2 > y^2.

3) The square root of both sides cannot be taken to obtain an answer.

Inequalities with an absolute value term will typically have two solutions. In order to simplify solving absolute value inequalities on the GMAT examination, candidates should follow these steps:

1)    Isolate the absolute value expression found in the inequality
2)    Set the expression so that it equals to both a positive and a negative value and
3)    Solve for the inequality.

Let’s take a look an inequality with only one absolute value term.
    
    5 +  | 2x + 1 | < 10

Let us follow the shortcuts

1) First, we will isolate the absolute term from the inequality, so that it is only on one side of the expression:

    5 – 5  + | 2x + 1 | < 10 – 5
   
    | 2x + 1 | < 5

2) Next, we will need to find both the positive and negative values of the term. It is important for candidates to remember that most of the absolute value inequalities will have two solutions. Those who fail to remember this will often choose the wrong answer. We can set the positive value by keeping the equation as it is; thus,

2x + 1 < 5

3) We can then apply basic algebraic manipulations to solve this question.

    2x + 1 – 1 < 5 – 1
    2x < 4
    x <2 --> 1 (Positive Solution)

By doing this, we will have the first answer required for solving the absolute value inequalities; however, we need not forget that there is a negative solution as well. To set up the negative solution, we make the absolute value inequalities negative; thus,

    - ( 2x + 1 ) < 5
    - 2x – 1 < 5
    - 2x – 1 + 1 < 5 + 1
    - 2x < 6
    
This is where the tricky part happens. Most candidates will forget to flip the inequality sign when they either divide or multiple by a negative number. Since in this case, we are multiplying 6 by a -2, we will need to apply this trick.

    X > -3 --> 2 (Negative Solution)

From Statements 1 and 2 we can assume that the value x is greater than -3, and smaller than 2 which can be written as -3 < x < 2.

The most recommended trick for solving multiple inequalities will be to solve each inequality separately and then combine them.

For example, let’s take a look at | x + 1 | >  3 and | x – 2 | < 4

To solve each inequality separately, we would look at | x + 1 | > 3 first. Just like before, it is important to get both the positive and the negative solutions.

For the positive solution of | x + 1 | > 3, we would get:

| x + 1 | > 3
x + 1 – 1 > 3 – 1
x > 2

And, for the negative solution of | x + 1 | > 3, we would get:

    | x + 1 | > 3
    - ( x + 1 ) > 3
    - x – 1 > 3
    -x – 1 + 1 > 3 + 1
    - x > 4
    x < -4

For the positive solution of | x – 2 | < 4, we would get:

    | x – 2 | < 4
    x – 2 + 2 < 4 + 2
    x < 6

For the negative solution of | x – 2 | < 4

  | x – 2 | < 4
- ( x – 2 ) < 4
- x + 2 < 4
- x + 2 – 2 < 4 – 2
- x < 2
x > -2

As we can see, the solutions for | x + 1 | > 3 and | x – 2 | <  4 are

x > 2,     x < -4 , x < 6 and x > -2

The overlapping regions will be 2 < x < 6

Solving inequalities with absolute value in GMAT is not too difficult once the correct theorems are applied. Candidates interested in performing well on the quantitative section of the GMAT examination should consider taking some time to practice the absolute term inequalities with only one inequality and then with multiple inequalities.