GRE Quantitative Algebra – Solving Linear Inequalities

Solving a question that contains only inequality, or only an equation that is specifically based on equations is rather a fun math problem to solve. However the GRE Quantitative Reasoning is here to test your complete mathematical skills and it is true they won’t be asking you straight-forward questions. They will mingle both together to confuse you. But with a little practice and inequalities will be your least troublesome topic in GRE, in fact if your concepts are clear not even a single point will be deducted in your exam.

So, what does Inequalities mean?

In simple words if spoken, any equation without the sign of equality is known as inequality.

Types of inequalities:

  • x greater than y, x >y
  • x less than y, x<y
  • x less than or equal to y, x ≤ y
  • x greater than or equal to y, x ≥y
  • x is not equal to y, x ≠ y
  • x lying in between range a1 and a2, a1 <x < a2; this is known as compound inequality.

The basic difference between an equation and an inequality is, an equation gives you the concrete result of a variable whereas an inequality only provides you with a range for the variable.

For example, consider these two equations x = 3 and x >3

The first equation gives you a single value, i.e. x = 3

While the second one is an inequality and it does not provide any specific value of x, in place it only restricts the value of x that it can’t be less than or equal to 3.

  • Addition and Subtraction of inequalities

Question: x + 4 > 16. Find x.

Solution: x + 4 > 16

-4 -4

x > 12

Let’s take another example:

Question: -p + 2q > q – 2p, which of the following holds good?

  1. p > 0
  2. q > 0
  3. p + q > 0
  4. p + q <0

Solution: Lets combine these terms together:

-p + 2q > q – 2p

+2p +2p


p + 2q >q


  • p + 2q > q

-q > -q

p + q >0

Hence the correct option is (3).

  • Multiplication and division of inequalities

Important Tip: If you are multiplying or dividing an inequality with a positive value then the arrow of inequality does not change but if the value to be multiplied or divided is negative then the arrow should be reversed.

For example: -6x > 12

Solution:-6x-6 >12-6

x >-2

But remember the tip given above, flip the arrow

x < -2

Another point to keep in mind is you cannot divide or multiply an inequality with a variable. The reason behind this rule is that, the sign of variable is unknown so it would be incorrect to divide or multiply by a variable.

The questions that are given in the GRE exam on the topic of inequalities is very small. The level of questions are also very basic, so it is possible to solve the questions in the test using what you’ve just learnt above.

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