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It’s safe to say that absolute value is a quant topic that many students dread. So, I’ll venture a guess that if you’ve found your way here, you’d like to learn more about GMAT absolute value. This article will delve into what absolute value is, how to calculate absolute value, and how to solve GMAT absolute value questions.
Here are the topics we’ll cover:
- What Do We Mean by Absolute Value?
- Expressing Absolute Value
- Solving Absolute Value Equations
- How to Deal With Two Absolute Values That Are Equal
- How to Solve Absolute Value Inequalities
- Key Takeaways
- What’s Next?
Before we jump into GMAT absolute value practice problems, let’s learn some basic absolute value concepts.
What Do We Mean by Absolute Value?
The absolute value of a number is the distance between that number and zero on the number line.
For example, we can say:
- The absolute value of 50 is 50 because it’s 50 units from zero.
- The absolute value of -50 is also 50 because it’s 50 units from zero.
We can display this on a number line.
With positive and negative numbers alike, the absolute value expression will always equal a positive number unless that number is zero.
TTP PRO TIP:
The absolute value of a number is the positive distance between that number and zero.
Next, let’s discuss absolute value notation.
Expressing Absolute Value
Using a number line, we saw in the previous section that the absolute value of a number is the positive distance between that number and zero. More formally, however, we express absolute value by the use of a specific notation: two vertical bars (called absolute value bars) with a value or an expression enclosed between them. For example, we express the absolute value of x as |x|.
Returning to our examples from the previous section, we can say:
- The absolute value of 50 can be expressed as |50| = 50.
- The absolute value of -50 can be expressed as |-50| = 50.
Notice that whether we are dealing with a positive number or a negative number, the absolute value is the same.
TTP PRO TIP:
We use absolute value notation, two vertical bars, to express the absolute value of numbers or variables.
Let’s practice interpreting absolute value notation with a basic absolute value GMAT question.
Example 1: Absolute Value – Basic Question
What is the complete solution set for |x| = 14?
- x = -14
- x = 14
- x = 0 and x = 14
- x = -14 and x = 0 and x =14
- x = -14 and x = 14
Solution:
The question stem is asking for all numbers that are 14 units from 0 on the number line. We see that both -14 and 14 satisfy this criterion.
Answer: E
While the above problem may seem basic, the main rule tested is used in countless GMAT absolute value questions.
Next, let’s discuss how to solve a more challenging absolute value equation.
Solving Absolute Value Equations
We now know some basic information about absolute value and the use of absolute value notation. However, what happens when we have an absolute value expression in a linear equation, or we have an algebraic expression enclosed within the absolute value bars? Although these problems look difficult, there are properties of absolute value equations we can follow to make solving these problems pretty darn easy! Once you learn these properties, be sure to summarize them on flashcards and commit them to memory.
When we are given absolute value equations, we generally solve for the variable inside the bars for two cases: positive and negative. Let’s work through a very simple example to illustrate this two-case procedure.
If |x + 1| = 12, what are the possible values of x?
To solve for x, we follow a two-case process. In the positive case, we drop the absolute value bars and solve the equation. In the negative case, we drop the absolute value bars and negate the entire expression inside the bars.
Case One: The Positive Case
We simply drop the absolute value bars and leave the expression untouched, giving us:
x + 1 = 12
x = 11
Case Two: The Negative Case
We drop the absolute value bars and multiply what was inside the absolute value bars by -1. Then, we solve for the variable, giving us:
-(x + 1) = 12
-x – 1 = 12
-x = 13
x = -13
So, x can be either 11 or -13.
TTP PRO TIP:
When we solve an absolute value equation, we always have a positive case and a negative case.
Let’s solve the following absolute value equation for x.
Example 2: Absolute Value – Solving an Equation
If |3p – 12| = 6, then what are the possible values of p?
- 6 and -6
- -2 and -6
- -2 and 2
- 2 and 6
- 6 only
Solution:
Let’s refine our nomenclature, making it a bit more descriptive.
Case One: 3p – 12 is positive (the positive case)
3p – 12 = 6
3p = 18
p = 6
Case Two: 3p – 12 is negative (the negative case)
-(3p – 12) = 6
-3p + 12 = 6
-3p = -6
p = 2
Thus, p is 2 or 6. One thing to remember is that, in addition to following the two-case process, we use our skills learned from solving linear equations.
Answer: D
TTP PRO TIP:
When we have an absolute value equation in which an algebraic expression is enclosed inside the absolute value bars, we use the “Case 1-Case 2” technique.
Let’s practice with another example.
Example 3: Absolute Value – Solving an Equation
What is the product of the solutions of |4x – 3| = 7?
- -5/2
- -3/2
- -1
- 3/2
- 5/2
Solution:
To determine the solution, we again use the two-case process.
Case One: 4x – 3 is positive
4x – 3 = 7
4x = 10
x = 10/4 = 5/2
Case Two: 4x – 3 is negative
-(4x – 3) = 7
-4x + 3 = 7
-4x = 4
x = -1
Now, we can calculate the product of the two solutions, which are 5/2 and -1.
5/2 x (-1) = -5/2
Answer: A
Let’s practice with another example, this time one that has an extra constant term in the equation and a coefficient in front of the absolute value expression.
Example 4: Absolute Value – Extra Constant and Coefficient
If 5|x +4| + 3 = 13, then x could be which of the following?
- 6
- 2
- -6/5
- -6
- -8
Solution:
While this equation may seem a bit more complicated than the previous ones we solved, it’s actually not. Our objective is to first isolate the absolute value expression on one side of the equation, and then use the two-case process.
5|x +4| + 3 = 13
First, we subtract 3 from both sides of the equal sign:
5|x +4| = 10
Next, we divide both sides by 5:
|x + 4| = 2
Now we are ready to follow the two-case process!
Case One: x + 4 is positive
x + 4 = 2
x = -2
Case Two: x + 4 is negative
-(x + 4) = 2
-x – 4 = 2
-x = 6
x = -6
So, x is -2 or -6. We see that only -6 is present among the answer choices.
Answer: D
TTP PRO TIP:
Before solving an absolute value problem, isolate the absolute value expression by performing appropriate algebraic procedures.
Let’s practice with one more example. In this example, the absolute value bars enclose a quadratic expression.
Example 5: Absolute Value – Quadratic Inside the Bars
If |w^2 – 3w – 7| = 3, what is the sum of all values of w that satisfy this absolute value equation?
- 10
- 6
- 0
- -2
- -4
Solution:
Let’s use the two-case process.
Case One: w^2 – 3w – 7 is positive
w^2 – 3w – 7 = 3
w^2 – 3w – 10 = 0
(w – 5)(w + 2) = 0
w = 5 OR w = -2
Case Two: w^2 – 3w – 7 is negative
-(w^2 – 3w – 7) = 3
Note that rather than distributing the -1 to the expression w^2 – 3w – 7, we divide both sides of the equation by -1
w^2 – 3w – 7 = -3
w^2 – 3w – 4 = 0
(w – 4)(w + 1) = 0
w = 4 OR w = -1
The sum of all values of w that satisfy the absolute value equation is 5 – 2 + 4 – 1 = 6.
Answer: B
Next, let’s discuss how to deal with two absolute values that are equal to each other.
How to Deal With Two Absolute Values That Are Equal
So far in this article about absolute value functions, we have dealt with absolute value equations in which we have one set of absolute value bars. However, you also may see an equation that has two sets of absolute values equal to each other. While again, this type of problem may seem daunting, we have another two-case procedure we can follow to solve these equations.
Case One: Drop the absolute value bars of both expressions.
We drop the absolute signs of both expressions and solve for the value of the variable.
Case Two: Drop the absolute value bars of both expressions and negate one of the expressions.
We distribute a negative into one of the absolute value expressions. Note that it doesn’t matter which one!
TTP PRO TIP:
For an equation in which two absolute value expressions are equal to each other, use a two-case process.
Let’s see how this process works with the following example.
Example 6: Absolute Value – Two Absolute Value Expressions
If |3n – 2| = |5n + 10|, then n could be equal to which of the following?
- -5
- -1
- 1
- 3/2
- 2
Solution:
The question stem presents us with two absolute value expressions that are equal to each other. Therefore, we will use the two-case procedure.
Case One: Drop the absolute value bars from both expressions, and then solve for the value of n.
3n – 2 = 5n + 10
-12 = 2n
-6 = n
Case Two: Drop the absolute value bars from both expressions, negate one of the expressions, and then solve for the value of n.
After you drop the absolute value bars from both expressions, choose either expression to distribute the negative. If we choose the expression on the right, |5n + 10|, we have:
3n – 2 = -(5n + 10)
3n – 2 = -5n – 10
8n = -8
n = -1
The value of n is -6 or -1.
Answer: B
Our final topic is absolute value inequalities.
How to Solve Absolute Value Inequalities
As we have often said, equations and inequalities share many properties. In other words, when you solve inequalities that contain absolute value expressions, the process is similar to when you solve equations of the same type, with one major difference. The difference is this: when you’re working the steps to solve the inequality, if you divide or multiply by a negative, you must reverse the inequality sign.
So, if you are up to speed on inequalities, solving absolute value inequalities should be a piece of cake!
TTP PRO TIP:
When solving absolute value inequalities, if you multiply or divide by a negative, you must reverse the inequality sign.
Absolute Value “Greater Than” Inequalities
Here is an example. Note that in this example, the inequality contains a “greater than” condition. This is an indication that we will have a union of two segments of the number line in our solution set, as indicated by the word “or” in our solution.
Example 7: Absolute Value “Greater Than” Inequality
Which of the following is not in the solution set of |3x – 11| > 7?
- -3/2
- 2/3
- 1
- 5/2
- 8
Solution:
We use the two-case procedure, as we have previously.
Case 1: 3x – 11 is positive
3x – 11 > 7
3x > 18
x > 6
Case 2: 3x – 11 is negative
-(3x – 11) > 7
-3x + 11 > 7
-3x > -4
Note that in order to solve for x, we must divide both sides of the equation by -3. Thus, we must reverse the inequality sign.
x < 4/3
The solution set includes all numbers greater than 6 OR all numbers less than 4/3. Thus, any number between 4/3 and 6 is not in the solution set. The number 5/2 satisfies this condition.
Answer: D
TTP PRO TIP:
Absolute value “greater than” inequalities yield a two-part “or” solution set.
Absolute Value “Less Than” Inequalities
In this next example, we encounter an absolute value inequality of the “less than” type. Our solution set will be the intersection of two inequalities, as indicated by the word “and” in our discussion of the solution.
Example 8: Absolute Value “Less Than” Inequality
Which of the following is in the solution set of the inequality |2x – 7| < 5?
- 0
- 1
- 3
- 6
- 8
Solution:
We again use the case 1-case 2 procedure.
Case 1: 2x – 7 is positive
2x – 7 < 5
2x < 12
x < 6
Case 2: 2x – 7 is negative
-(2x – 7) < 5
-2x + 7 < 5
-2x < -2
At this point, we divide both sides of the equation by -2, so we must reverse the inequality sign.
x > 1
The two inequalities yield x < 6 AND x > 1. Thus, the solution set is 1 < x < 6. Any value between 1 and 6 (not including 1 or 6) will be in the solution set. Of the answer choices, only 3 is in the solution set.
Answer: C
TTP PRO TIP:
Absolute value “less than” inequalities yield a single-interval solution set.
Key Takeaways
The absolute value of a number is that number’s distance from 0 on the number line. This definition is a simple one. However, when absolute value is used in an equation, solving the equation may be challenging.
Use the “Case 1-Case 2” procedure to solve an absolute value equation. In either case 1 or case 2, drop the absolute value bars. For Case 1, first drop the absolute value bars, and then solve the equation. For Case 2, drop the absolute value bars, and then negate the expression inside the (dropped) absolute value bars.
If you encounter two absolute value expressions set equal to each other, again use a “Case 1-Case 2” procedure. For Case 1, drop the absolute value bars from both expressions, and then solve the equation. For Case 2, again drop both sets of absolute value bars, but then negate one of the absolute value expressions.
To solve absolute value inequalities, again use the two-case procedure. For a “greater than” absolute value inequality, you will obtain two “or” inequalities that will define the solution set. For a “less than” absolute value inequality, you will obtain two “and” inequalities that will define the solution set.
What’s Next?
Absolute value is just one of many math topics tested on the GMAT. Mastering each topic is critical, and it is also important to become skilled in answering any question you might encounter. You can get a great start on your GMAT practice by taking a free practice GMAT exam. Then, read our expert tips on increasing your GMAT quant score.
Best of luck on your exam!
