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.

Example of Adding & Subtracting Inequalities

Example of Multiplying & Dividing Inequalities

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:

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).

More Information About Inequalities:
multiplication & division
addition & subtraction
absolute value
exponents

Technique: Boundary Testing

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

   Correct Answer: C

   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)	2<x<7
   B)	x>6 and x<0
   C)	x>10 and x<-2
   D)	3<x<6
   E)	x>3 and x<1

   Correct Answer: D

   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

   Correct Answer: B

   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