# Inequalities - GMAT Math Study Guide

## Definitions

An inequality compares two statements with different values.

• Inequality - A comparison of two values or expressions.
For example, x > 10 is an inequality whereas x = 10 is an equation.
• Equation - A statement declaring the equality of two expressions.
For example, 5x + 5 = 30 is an equation whereas 5x + 5 < 30 is an inequality.

## Five Types of Inequalities

### Greater Than (>) With a greater than inequality, all numbers larger than x (but not equal to x) fit the inequality. For example, any number larger than 7 fits the above inequality (e.g., 7.1, 7.2, 8, 9, 10, ...).

### Less Than (<) With a less than inequality, all numbers smaller than x (but not equal to x) fit the inequality. For example, any number smaller than 7 fits the above inequality (e.g., 6.9, 6.8, 6, 5, 0, -1, -100, ...)

### Greater Than or Equal to (>) With a greater than or equal to inequality, all numbers greater than x (and equal to x) fit the inequality. For example, any number that is 7 or is larger than 7 fits the above inequality (e.g., 7, 7.1, 7.2, 8, 9, 10, ...).

### Less Than or Equal to (<) With a less than or equal to inequality, all numbers smaller than x (and equal to x) fit the inequality. For example, any number that is 7 or is smaller than 7 fits the above inequality (e.g., 7, 6.9, 6.8, 6.5, 6, 5, 0, -1, -100, ...).

### Not Equal to (≠) With a not equal to inequality, any number that is not equal to the number in question (7 in this case) fits the inequality. For example, any number that is not 7 fits the above inequality (e.g., 7.1, 6.9, 6, 8, 10, 0, -100, ...).

## Operations With Inequalities

When one action (e.g., addition, multiplication, subtraction, or division) is done to one side of the inequality, the same action must be done to the other side of the inequality. However, there is one major catch: if both sides of an inequality are multiplied or divided by a negative number, then the inequality sign must be flipped.

If x+2 < y+2 is multiplied by -1 on both sides, the result is not -2-x < -y-2. Instead, the result is -2-x > -y-2.

### Example of Adding & Subtracting Inequalities

5x - 7 < 4x + 3
5x - 7 + 7 < 4x + 3 + 7
5x < 4x + 10
5x - 4x < 4x - 4x + 10
x < 10

### Example of Multiplying & Dividing Inequalities

15x < 30
x < 2

Since dividing or multiplying by a negative number flips the inequality sign, you cannot multiply or divide by an unknown (i.e., a variable), as it could be negative. This is a common trap. For example:

xz < 10z cannot be solved by dividing both sides by z to get x < 10.
If z were negative, the inequality would end up as x > 10. Unless a problem states that a variable is positive or negative, both sides cannot be divided or multiplied by an unknown as you cannot be certain whether to flip the inequality sign.

It is extremely important that you cement this property of inequalities into your mind as the test makers like to trick people on this topic (more information about this important topic within multiplying & dividing inequalities).

multiplication & division
absolute value
exponents

## Technique: Boundary Testing

If 2 < x - 6 < 10 and 25 < y + 10 < 45, what inequality represents the range of values of x + y?

1.) Solve each inequality separately.
2 < x - 6 < 10
2 + 6 < x - 6 + 6 < 10 + 6
8 < x < 16

25 < y + 10 < 45
25 - 10 < y + 10 - 10 < 45 - 10
15 < y < 35

2.) Combine each inequality by using the boundary of each inequality to find the end of the combined (i.e., summed, x + y) inequality.

2a.) Find the smallest possible value of the inequality.
In the first inequality: x is 8.000...0001
In the second inequality: y is 15
23 < x + y

2b.) Find the largest possible value of the inequality.
In the first inequality: x is 16
In the second inequality: y is 34.9999...
x + y < 51
3.) Combine each value from step 2 to find the inequality that encapsulates x + y.

3a.) Find the smallest possible value of the combined inequality.
8.000...0001 + 15 produces x + y > 23

3b.) Find the largest possible value of the combined inequality.
16 + 34.9999 produces y < 51

Putting these together: 23 < x + y < 51

## Types of GMAT Problems

1. Solving Inequalities

Like equations, inequalities are solved by isolating the unknown by performing the same operations to both sides.

Solve for x: 4 - 2x<10
 A) x<3 B) x<-5 C) x>-3 D) x>5 E) x<6
1. There are a few different algebraic ways to solve this problem.
2. Method 1
1. Isolate x on one side.
4-2x<10
-2x<6 (subtract 4 from both sides)
x>-3 (divide both sides by -2 and flip the inequality)
3. Method 2
1. Isolate x on one side.
4-2x < 10
4 < 2x+10 (add 2x to both sides)
-6 < 2x (subtract 10 from both sides)
-3 < x (divide both sides by 2)
2. Combining Inequalities

In order to combine a group of inequalities, each one must be solved (unknown isolated). Then, each inequality must be rearranged so the inequalities are all pointing in the same way, preferably less than. The next step is to line up the common unknown. The last step is to combine the inequalities using the most restricting upper and lower extremes.

What is the range of possible x values given:
2x+8<20
5x>15
 A) 26 and x<0 C) x>10 and x<-2 D) 33 and x<1
1. Solve both equations for x.
2x+8<20
2x<12 (subtract 8 from both sides)
x<6 (divide both sides by 2)

5x>15
x>3 (divide both sides by 5)
2. Rewrite the second inequality so it is in the less than format:
3<x
3. List the two inequalities so the x variable lines up and then combine them. 3. Inequalities Involving Unknowns of Power Two

Like equations, inequalities involving powers of 2 can have up to two solutions. To find all the answers, isolate the unknown. Since the unknown is squared, take the square root of both sides. However, when taking the square root of the unknown, both the positive and negative square root of the other side must be taken. Two inequalities will be formed, one with a positive square root and the other with a negative square root and a flipped inequality. Both inequalities can be solved for different parts of the solution.

x2 + 9 < 34
 A) x<6 and x>-6 B) x<5 and x>-5 C) x>5 and x<-5 D) x>6 and x<-6 E) x<3 and x>-3
1. Isolate x.
x2 + 9 < 34
x2 < 25 (subtracted 9 from both sides)
2. Remember that due to the even exponent, x could be a negative number.
Take the square root of both sides. In taking the square root, there are two possibilities.
Case (1) is that X is positive:
x2 < 25
x<5

Case (2) is that X is negative:
x2 < 25
Since we are raising both sides by 1/2 and the number x is negative in this case, we must flip the inequality sign:
x>-5
3. Combine the two inequalities:
-5 < x < 5
4. As a check, you could try a few numbers:
x = 4 yields 25, which is less than 34
x = -4 yields 25, which is less than 34
But:
x = 6 yields 45, which is not less than 34
x = -6 yields 45, which is not less than 34