GRE Exam: A Simple Guide To Algebra - Inequalities and Absolute Value

Inequalities look like the following:

 

Any time you see <, >, ≤, ≥, you are dealing with an inequality.

The following list translates inequalities:

Types of Inequalities
- a> b means “a is greater than b”
- a< b means “a is less than b”
- a≥ b means “the value of a is at least equal to the value of b”
- a≤ b means “the value of a is at most equal to the value of b”
- 3 < a< 5 means “the value of a is between 3 and 5” (This is called a compound inequality.)

Inequalities Versus Equations
The fundamental difference between equations and inequalities is the following: Whereas an equation will give you a concrete value for a variable, an inequality will only give you a range.

Compare:

If you plot these on the number line, you will see that the equation x = 7 provides one and only one value for x.

On the other hand, the inequality x > 7 does not provide a specific value; instead, it restricts the possible values that x can be. Since x > 7, it can only be any number to the right of 7 on the number line.

Manipulating Inequalities
As is the case with equations, your initial goal with inequalities will usually be to simplify what’s given to you. Fortunately, most of the rules you have learned for manipulating equations will also apply to inequalities.

Addition and Subtraction with Inequalities
 

x + 3 > 12. Solve for x.
SOLUTION:


Here is another example:
If −x + 2y > y − 2x, then which of the following must be true? (Indicate all that apply.) (A) x > 0 (B) y > 0 (C) x + y > 0

SOLUTION: Combine like terms:

The correct answer is C.

Multiplication and Division with Inequalities
When multiplying or dividing across an inequality, keep in mind the following rules:

If you multiply or divide across an inequality by a positive value, the inequality arrow does not change.

Example: If 2x > 6, what is the range for x?

SOLUTION: To isolate x, divide both sides of the inequality by 2.

Note that the sign does not change, since you are dividing by a positive.

If you multiply or divide across an inequality by a negative value, the inequality arrow flips.
 

Example: −2x < 6. Solve for x.

SOLUTION: Divide both sides by −2:

But remember to flip the sign: x > −3.
You cannot multiply or divide across an inequality by an unknown.

For example, if you are told that , you may be tempted to multiply both sides by y and arrive at x > y.

However, this would be incorrect. Why? Because you do not know the sign of y. Since you don’t know the sign of y, you do not know whether the inequality arrow will flip when you multiply.

Thus you need to keep the inequality in its original form.

Manipulating Compound Inequalities
From the introduction to this section, recall that a compound inequality looks like the following: −7 < a + 3 < 12.

The rules for manipulating compound inequalities are the exact same ones as those for normal inequalities. Just make sure that you perform the same operation on all three parts of the inequality.

Let’s solve for a in the preceding example:


Extremes with Inequalities
In some inequality questions, you will be presented with multiple inequalities, or with an inequality and an equation, and will be asked to draw inferences about their products.
In these examples, choosing extreme values for the variables is often the optimal approach.

Example: If a = 3 and −6 < b < 12, which of the following can equal ab? (Indicate all that apply.) (A) –18 (B) 0 (C) 18 (D) 24 (E) 36

SOLUTION: Since you are trying to figure out possible values of ab, you should consider the greatest value that ab could be and the smallest value that ab could be.
Since you know a = 3, the product will be smallest when b is smallest. So choose the extreme value for b: in this case, −6. If b = −6, then ab = −18.
However, you know b > −6. Therefore, ab > −18. Now try the upper bound. If b = 12, then ab = 36. However, you know that b < 12, meaning that ab < 36. You arrive at the compound inequality: −18 < ab < 36. The answer is B, C, and D.

Maximization and Minimization with Inequalities
Another common type of inequality question will give you two inequalities and ask you for the maximum or minimum value of their product. In these cases, it is essential to consider the extremes for all variables.

Example: If −7 ≤ a ≤ 12 and −11 ≤ b ≤ 5, what is the maximum value of ab?
SOLUTION: The trap here is to multiply the maximum value for a and the maximum value for b and arrive at 60. However, note that if a and b are both negative, their product will be positive!

Thus it is possible that the product of the smallest values of a and b will yield a larger value than the product of the largest values of a and b. And that is exactly what happens here: take the minimum value for a, −7; and the minimum value for b, −11; and the product is 77, which is greater than 60.

Let’s look at an example with minimization:
If −12 ≤ q ≤ 9 and 8 ≤ r ≤ 12, then the minimum value of qr = ?

SOLUTION: As in the preceding example, it might be tempting to multiply the smallest value for q and r and arrive at −96. However, note that when you multiply a negative and positive value, the result becomes smaller as the positive number becomes larger—for example, −3(9) < −3(7). Thus you will minimize qr when you multiply the minimum value of q by the maximum value of r: −12 × 12 = −144.

Absolute Value
In its simplest form, absolute value refers to the distance between a number, variable, or expression and zero. Absolute value is denoted using brackets, for example, |x + y| or |−3|.
Since absolute value refers to distance, the absolute value of a number or expression will always be positive. For example, |−8| = 8 since −8 is 8 units away from zero.

Absolute Value with Unknowns
When an unknown term or expression is within an absolute value, there will be two possible values for the unknown. For example, if |x| = 2, then x = 2 or x = −2.

Why are there two values for x? Because absolute value refers to distance! Both 2 and −2 are two units away from zero, so x can equal either of those values.

When solving for an unknown within an absolute value, use the following process:
Example: If 9 + |x + 4| = 28, what are the possible values for x?

Step 1: Isolate the absolute value:

Step 2: Create two equations. In one equation, the expression inside the absolute value will equal the positive value on the right.

In the other equation, the expression inside the absolute value will equal the negative version of the value on the right:

Step 3: Solve for the unknown in both equations:


 

Absolute Values and Inequalities
In tougher absolute value questions, you will be given a range for the absolute value instead of a concrete value.

For example:


To solve these questions, take the following approach:

Example: If |x + 3| < 7, what of the following describes the range for x?

Step 1: Set up two solutions:

Step 2: Multiply by −1 (flip the sign!):

Step 3: Combine the inequalities:


Test Positives and Negatives
When answering a “must be true” question or Quantitative Comparison question with absolute values, it is helpful to test positive and negative cases.

For this question, indicate all of the answer choices that apply.
Example: If x ≠ 0, then which of the following must be true? (Indicate all that apply). (A) |x| = x (B) (C) (D) (E)

SOLUTION: Choose a positive and a negative value for x, and see which choices are true for both cases. Let’s use −2 and 2 for x.

Note that you will start with the negative case, since this is the case most likely to contradict the given equations.

A: |−2| = 2. 2 ≠ −2     → Eliminate Choice A.
B: .

The equation is true when x = −2.

Now try .

The equation is true when x = −2. → Keep Choice B.
C:
D:
E: |−2| × |−2| = 2 × 2 = 4. (−2)2 = 4.

The equation is true when x = −2.

Now try 2: |2| × |2| = 2 × 2 = 4. (2)2 = 4.

The equation is true when x = 2. → Keep Choice E.

The correct answer is B and E.