Which Of The Following Values Are In The Range

Which of the Following Values Are in the Range? A Step‑by‑Step Guide

Understanding whether a given number falls inside a specified interval is a fundamental skill in mathematics, statistics, and everyday problem‑solving. This article walks you through the concept of ranges, shows you how to test values systematically, and provides plenty of examples so you can confidently answer questions like “which of the following values are in the range?”

Introduction

When a problem presents a set of numbers and asks you to pick those that lie within a certain range, it is really asking you to apply a simple inequality test. The range can be expressed in several ways—using words (“between 5 and 10”), symbols (“[5, 10]”), or a combination of inclusive and exclusive endpoints. Mastering the technique not only helps you ace homework and exams but also sharpens logical thinking for real‑world situations such as budgeting, data filtering, and quality control.

Below, we break down the process into clear steps, illustrate each with varied examples, highlight common pitfalls, and finish with practice questions and a FAQ section.

Understanding What a Range Means

A range (also called an interval) defines all real numbers that satisfy a lower bound and an upper bound. The bounds can be:

The key idea is to replace the verbal description with the appropriate inequality and then test each candidate value against it. ---

Steps to Determine If a Value Lies in a Range

Follow these four steps for any value‑checking task:

1. Identify the bounds – Write down the lower limit (L) and the upper limit (U) exactly as they appear (including whether they are inclusive or exclusive).
2. Translate to inequalities – Convert the interval notation into one or two inequality statements.
3. Test each candidate – Plug the value into the inequality(ies). If all conditions are satisfied, the value is inside the range; otherwise, it is outside.
4. Record the result – List the values that passed the test.

Let’s apply these steps to a few typical scenarios.

Example 1: Simple Integer Range

Problem: Which of the following values are in the range ([3, 8])?
Values: ({2, 3, 5, 7, 9, 8})

Solution:

1. Bounds: (L = 3) (inclusive), (U = 8) (inclusive).
2. Inequalities: (3 \le x \le 8).
3. Test each value:

4. Result: The values inside the range are 3, 5, 7, 8.

Example 2: Exclusive Upper Bound Problem: Which of the following values are in the range ((0, 10))?

Values: ({0, 0.5, 5, 9.9, 10, 10.1})

Solution:

1. Bounds: (L = 0) (exclusive), (U = 10) (exclusive).
2. Inequalities: (0 < x < 10).
3. Test:

4. Result: The values inside the range are 0.5, 5, 9.9.

Example 3: Mixed Inclusivity

Problem: Which of the following values are in the range ([-4, 2))?
Values: ({-5, -4, -3.5, 0, 1.9, 2, 2.1})

Solution:

1. Bounds: Lower (-4) (inclusive), Upper (2) (exclusive).
2. Inequalities: (-4 \le x < 2). 3. Test:

4. Result: The values inside the range are -4, -3.5, 0, 1.9.

Example 4: Unbounded Range Problem: Which of the following values are in the

range ((-\infty, 5])?
Values: ({-10, 0, 5, 5.1, 10})

Solution:

1. Bounds: Lower unbounded (no lower limit), Upper (5) (inclusive).
2. Inequality: (x \le 5).
3. Test:

4. Result: The values inside the range are -10, 0, 5.

Example 5: Two-Sided Exclusive Range

Problem: Which of the following values are in the range ((1, 4))?
Values: ({0, 1, 2, 3.9, 4, 5})

Solution:

1. Bounds: Lower (1) (exclusive), Upper (4) (exclusive).
2. Inequality: (1 < x < 4).
3. Test:

4. Result: The values inside the range are 2, 3.9.

Conclusion

Determining which values fall within a given range is a straightforward process once you clearly identify the bounds and their inclusivity. By translating the range into inequalities and systematically testing each value, you can quickly and accurately separate those that belong from those that do not. This method works for simple integer intervals, mixed inclusivity, unbounded ranges, and even more complex scenarios. Mastering this skill is essential for data validation, problem-solving, and many real-world applications where precise value filtering is required.