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It’s safe to say that absolute value is a GRE quant topic that most students dread. So, I’ll venture a guess that if you’ve found your way here, you’d like to learn more about absolute value on the GRE. This article will dive into what absolute value is, how to calculate absolute value, and how to solve absolute value equations.
Here are all the topics we’ll cover:
- What Is Absolute Value?
- How to Express Absolute Value
- How to Solve Absolute Value Equations
- Absolute Value Equations Containing Multiple Terms
- How to Deal With Two Absolute Values That Are Equal
- How to Solve Absolute Value Inequalities
- Summary
- What’s Next?
Before jumping into solving absolute value questions, let’s discuss some basic absolute value concepts.
What Is Absolute Value?
The absolute value of a number is the distance between that number and zero. Generally, this can be expressed 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 50 because it’s 50 units from zero.
We can express this on a number line.
Whether the number in question is positive or negative, 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.
How to Express Absolute Value
Using a number line, we showed in the previous section how the absolute value of a number is the positive distance between that number and zero. However, in math, we express absolute value using a specific notation, two vertical bars (or absolute value bars) with a value or an expression between them. So, we can express the absolute value of x as |x|.
Referring 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.
TTP PRO TIP:
We express the absolute value of numbers or values using absolute value notation.
Let’s practice using absolute value notation with the following quantitative comparison question.
Absolute Value Example 1:
n < 0 and p > 0
Quantity A:
|n|
Quantity B:
p
- Quantity A is greater.
- Quantity B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.
Solution:
Although this problem is actually pretty straightforward, watch out for the trap! In the given information, we see that n is a negative number and p is positive.
However, in quantity A, we are given the absolute value of n, which of course must be positive, and we also know that p is positive.
However, without knowing the values of either n or p, we cannot determine which quantity is greater.
Answer: D
Next, let’s discuss solving absolute value equations.
How to Solve Absolute Value Equations
We now know some basic information about absolute value and how to express absolute values mathematically. However, what happens when we have an absolute value in an equation or an algebraic expression enclosed within the absolute value bars? This may seem scary, but I promise, there are specific properties of absolute value equations we can follow to make solving these problems pretty darn easy! Once you learn the process, make sure to put it on a flashcard, so you can get it memorized!
Any time we have an absolute value equation, to solve for what’s inside the absolute value bars, we solve for whatever is inside of the bars for two cases: positive and negative. Let’s work through a very simple example to illustrate.
If |x| = 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 expression that was inside the bars.
Case One: Positive Value
We drop the absolute value bars and leave the variable untouched, giving us:
x = 12
Case Two: Negative Value
We drop the absolute value bars and multiply what was inside the absolute value bars by -1, giving us:
-x = 12
x = -12
So, x can be either 12 or -12.
You may be thinking, did I have to go through all these steps for a simple question? The short answer is no! Rather, you could have said x is 12 or -12 without doing our two-case process. But, when the equations become more complicated, the two-case process will be necessary.
TTP PRO TIP:
When we have |x| = n, x equals positive or negative n.
Let’s practice with an example.
Absolute Value Example 2
If |n| = 5 and |m| = 4, then n + m could be which of the following?
Indicate all that apply.
- -9
- -5
- -1
- 1
- 5
- 9
Solution:
Rather than using the two-case method, we use the fact that when |n| = 5, n is 5 or -5, and when |m| = 4, m is 4 or -4.
So, let’s now calculate the possible sums of n and m.
When n = 5 and m = 4:
5 + 4 = 9
When n = 5 and m = -4:
5 + (-4) = 1
When n = -5 and m = 4:
-5 + 4 = -1
When n = -5 and m = -4:
-5 + (-4) = -9
Thus, the sum of n and m can be -9, 1, -1, and 9.
Answer: A, C, D, F
Next, let’s look at some more complicated absolute value equations.
Absolute Value Equations Containing Multiple Terms
In the previous section, we discussed dealing with an absolute value equation with a single term. While we decided that we did not have to follow the laborious two-case process for single terms in the absolute value, we should follow it when we have additional terms inside the absolute value bars. For example:
If |3p – 18| = 12, then what are the possible values of p?
Now is the right time to follow the two-case process!
Case One: 3p – 18 is positive
3p – 18 = 12
3p = 30
p = 10
Case Two: 3p – 18 is negative
-(3p – 18) = 12
-3p + 18 = 12
-3p = -6
p = 2
Thus, p is 2 or 10. One thing to keep in mind is that, in addition to following the two-case process, we use our skills learned from dealing with linear equations.
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 an example.
Absolute Value Example 3
|4n – 10| = 20
Quantity A:
n^2
Quantity B:
n^3
- Quantity A is greater.
- Quantity B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.
Solution:
To simplify the given information, we can use the two-case process.
Case One: 4n – 10 is positive
4n – 10 = 20
4n = 30
n = 30/4 = 15/2 = 7.5
Case Two: 4n – 10 is negative
-(4n – 10) = 20
-4n + 10 = 20
-4n = 10
n = -10/4 = -5/2 = -2.5
Now, we can evaluate the quantities.
When n = 7.5, n^2 = 56.25 and n^3 = 421.875. Thus, n^2 < n^3.
When n = -2.5, n^2 = 6.25 and n^3 = -15.625. Thus, n^2 > n^3.
Thus, we cannot determine a relationship, and the answer is D.
Answer: D
Let’s practice another example, this time with an extra constant term in the equation and a coefficient in front of the absolute value expression.
Absolute Value Example 4
If 5 + 2|3x + 12| = 29, then x equals which of the following?
- -8
- -4
- -2
- 4
- 8
Solution:
While this equation may seem more complicated than the previous ones we solved, it’s not that difficult. That said, our goal is to isolate the absolute value term, and then follow the two-case process. Let’s do so now.
5 + 2|3x + 12| = 29
First, we subtract 5 from both sides:
2|3x + 12| = 24
Next, we divide both sides by 2:
|3x + 12| = 12
Now we are ready to follow the two-case process!
Case One: 3x + 12 is positive
3x + 12 = 12
3x = 0
x = 0
Case Two: 3x + 12 is negative
-(3x + 12) = 12
-3x – 12 = 12
-3x = 24
x = -8
So, x is 0 or -8. We see that only -8 is present among the answer choices.
Answer: A
Let’s practice one more, this time an absolute value question with a quadratic equation.
Absolute Value Example 5
If |x^2 – 3x – 7| = 3, then what is the sum of the possible values of x?
- -4
- -2
- 0
- 6
- 10
Solution:
Let’s follow the two-case process.
Case One: x^2 – 3x – 7 is positive
x^2 – 3x – 7 = 3
x^2 – 3x – 10 = 0
(x – 5)(x + 2) = 0
x = 5 OR x = -2
Case Two: x^2 – 3x – 7 is negative
-(x^2 – 3x – 7) = 3 (note that we divide by -1 here)
x^2 – 3x – 7 = -3
x^2 – 3x – 4 = 0
(x – 4)(x + 1) = 0
x = 4 OR x = -1
Therefore, the sum of all roots of the absolute value equation is 5 – 2 + 4 – 1 = 6.
Answer: D
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, we have dealt with absolute value equations in which we have one set of absolute value bars. However, you may also see an equation that has two sets of absolute values equal to each other. While again, this may seem scary, we have another two-case process we can follow to solve these equations.
Case One: Drop the absolute value bars.
We drop the absolute signs of both expressions and solve for the value of the variable.
Case Two: Drop the absolute value bars and negate one of the expressions.
We distribute a negative into one of the absolute value equations. Remember, it does not matter which one! So, choose one and don’t look back!
TTP PRO TIP:
When we have an equation in which two absolute value expressions are equal to each other, we solve using a two-case process: for Case One, we drop the absolute value signs and solve; for Case Two, we drop the absolute value bars and negate one of the expressions, and then we solve for the variable.
Let’s see how this process works with the following example.
Absolute Value Example 6
If |4n – 1| = |6n + 3|, then n could be equal to which of the following?
- -10
- -4
- -2
- 1
- 2
Solution:
We have two absolute value expressions equal to each other, so we can follow our two-case process.
Case One: Drop the absolute value bars and solve for the variable.
Let’s drop the absolute bars and solve:
4n – 1 = 6n + 3
-2n = 4
n = -2
Case Two: Drop the absolute value bars, negate one of the expressions, and then solve for the variable.
First, we drop the absolute value bars. Then, we can choose either expression to distribute a negative. So, let’s choose |4n – 1|, and we will leave the other expression alone. So, we have:
-(4n – 1) = 6n + 3
-4n + 1 = 6n + 3
-4n = 6n + 2
-2 = 10n
-1/5 = n
Thus, we see that n is -2 or -1/5, and since -2 is among the answer choices, it’s correct.
Answer: C
Let’s finish our discussion with absolute value inequalities.
How to Solve Absolute Value Inequalities
I have some good news! Solving inequalities with absolute value is quite similar to solving equations with absolute value, with one major difference. That difference is that, when solving the inequality, if you divide or multiply by a negative, you must flip 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:
Solving absolute value inequalities is similar to solving absolute value equations.
Let’s practice with an example.
Absolute Value Example 7
|2x – 8| > 12
Quantity A:
x
Quantity B:
1
- Quantity A is greater.
- Quantity B is greater.
- The two quantities are equal.
- The relationship cannot be determined from the information given.
Solution:
We again can follow our two-step process.
Case One: Solve for 2x – 8 as positive
We drop the absolute value bars, and then solve for x:
2x – 8 > 12
2x > 20
x > 10
Case Two: Drop the absolute value bars and negate the expression
We distribute the negative sign after dropping the absolute value bars:
-(2x – 8) > 12
-2x + 8 > 12
-2x > 4
Since we are dividing by negative 2, we flip the inequality sign, giving us:
x < -2
Thus, x < -2 or x > 10.
Therefore, there is no way to know whether x is less than or greater than 1.
Answer: D
Summary
The absolute value of a number is that number’s distance from 0 on the number line. This definition is simple enough, but when absolute value is used in an equation, you might have difficulty solving the equation.
A basic way to solve an absolute value equation is to use the “Case 1 – Case 2” technique. In either case, you drop the absolute value bars. Then, you solve the equation by first using the positive value of the expression inside the (dropped) absolute value bars. Then you solve the equation by using the negative value of the expression inside the (dropped) absolute value bars.
In the most challenging situation, you might encounter two absolute value expressions set equal to each other. In this scenario, you still will use the “Case 1 – Cae 2” technique. For Case 1, you first drop all absolute value bars, and then solve the equation. For Case 2, you drop both sets of absolute value bars, and then negate one of the absolute value expressions.
What’s Next?
Absolute value is just one of dozens of major math topics tested on the GRE. Learning all the topics is important, and it is also important to become proficient in answering any question the test can throw at you. Read this article to get some expert tips on increasing your GRE quant score.
Good luck!
