How to Solve an Inequality Between Two Numbers (With Examples)

Updated on July 22, 2020
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Mark, a math enthusiast, loves writing tutorials for stumped students and those who need to brush up on their math skills.

Learn how to find the set of integers that satisfies an inequality.
Learn how to find the set of integers that satisfies an inequality. | Source

If you're reading this, you're probably looking for some clarity about how to find all of the integers (whole numbers) that satisfy an inequality between two numbers. Perhaps you've been presented with a problem that looks something like this:

-2 ≤ X < 3

With an inequality like this, we need to find all possible values of X, our variable. Before we dive in, it's important to make sure we are familiar with all of the elements of this sort of problem. Let's start by defining a few terms and symbols.

Terms and Symbols to Be Familiar With

  • Integer: An integer is any whole number. This includes positive whole numbers (like 1, 2 and 3), negative whole numbers (like -1, -2 and -3), and zero (0).
  • Positive Integer: A positive integer is any whole number greater than 0 (like 1, 2, 3 and so on).
  • Negative Integer: A negative integer is any whole number less than 0 (like -1, -2, -3 and so on). Negative integers are preceded by the symbol "-" so that they can be distinguished from positive integers
  • X: X is the symbol we use as a variable, or placeholder for our solution. In the case of inequalities, X usually represents a series of numbers rather than a single number
  • <: This symbol means "less than" and is used to indicate that the number to its left (the pointy side) is less than the number to its right (the open side).
  • >: This symbol means "greater than" and is used to indicate that the number to its left (the open side) is greater than the number to its right (the pointy side).
  • ≤: This symbol means "less than or equal to" and is used to indicate that the number to its left (the pointy side) is less than or equal to the number to its right (the open side).
  • ≥: This symbol means "greater than or equal to" and is used to indicate that the number to its left (the open side) is greater than or equal to the number to its right (the pointy side).

How to Find All of the Integers That Satisfy an Inequality

Now that we're familiar with all of our terms and symbols, let's take another look at the example given above. We want to find a set of numbers that is a solution to:

-2 ≤ X < 3

In this case, X represents the set of numbers that will be our solution. Using what we learned above, let's translate the problem into words. We want to list a set of numbers that includes all integers that are greater than or equal to -2 and less than negative 3. We can visualize this set of numbers by thinking of them as if they exist on a line. Take a look at the image below.

-2 ≤ X < 3
-2 ≤ X < 3

The red line in the image above represents the set of numbers that satisfies our inequality. The circle above -2 is filled in because -2 is included in our set. The circle above 3 is not filled in because 3 is not included in our set. This is because our set includes all numbers greater than or equal to -2 (denoted by the ≤ symbol) and lesser than but not equal to (denoted by the < symbol) 3.

Knowing this, we can now confidently list out the integers that satisfy this inequality by counting up from -2 to the last integer before 3. The solution to -2 ≤ X < 3 is -2, -1, 0, 1 and 2.

Another Explanation With a New Example

If you are asked to write down all the integers that satisfy the inequality -3 < X ≤ 4, then you are looking for all of the values of X that are greater than -3 and less than or equal to 4. This is because -3 < X means X > -3 (X is more than -3) and X ≤ 4 means X is less than or equal to 4.

Since integers are whole numbers, you don’t need to write down any decimals or fractions. So, the integers that satisfy -3 < X ≤ 4 are -2,-1, 0, 1, 2, 3 and 4.

Example Problems With Solutions

Problem 1: Write down all of the integers that satisfy the inequality -2 ≤ X < 3.

Explanation: Here, -2 ≤ X means X ≥ -2, so you want to list all integers that are greater than or equal to -2. X < 3 means all integers less than 3.

Solution: The required integers are -2,-1, 0, 1 and 2.

Problem 2: Write down all of the integers that satisfy -4 < X < 2.

Explanation: Here, -4 < X means that X > -4, so we want to list all integers that are greater than -4 but less than 2.

Solution: The required integers are -3,-2, -1, 0 and 1.

Problem 3: Write down all of the integers that satisfy -6 ≤ 2X ≤ 5

Explanation: This time, we have 2X in the centre of the inequality, so the first thing we need to do is divide everything by 2 to isolate our variable. This gives us -3 ≤ X ≤ 2.5

-3 ≤ X is the same as X ≥ -3, so we want all integers greater than or equal to -3. X ≤ 2.5 means we want all integers less than or equal to 2.5 (don’t include 2.5 in your solution, as 2.5 is not an integer).

Solution: The required integers are -3, -2, -1, 0, 1 and 2.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

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