Solving the Inequality -1 ≤ 5 - 3x ≤ 20: A Comprehensive Guide

Understanding and solving inequalities is a fundamental skill in algebra. This guide will walk you through the steps to solve the inequality -1 ≤ 5 - 3x ≤ 20. We'll break down the process step-by-step, providing clear explanations and sample calculations to ensure a thorough understanding of the concept.

Step-by-Step Solution

Our goal is to solve the inequality -1 ≤ 5 - 3x ≤ 20. This involves isolating the variable x within the given boundaries. Here are the detailed steps:

Step 1: Isolate the x Term

The first step is to subtract 5 from all parts of the inequality to simplify the expression and isolate the term with x.

-1 - 5 ≤ 5 - 3x - 5 ≤ 20 - 5

This simplifies to:

-6 ≤ -3x ≤ 15

Step 2: Isolate x

The next step is to divide each part of the inequality by -3. Remember, when dividing by a negative number, the inequality sign flips.

-6/-3 ≥ -3x/-3 ≤ 15/-3

This simplifies to:

2 ≥ x ≤ -5

Or, equivalently:

2 ≥ x or x ≤ -5

Graphical Representation

Additionally, you can solve this inequality by graphing. Plot the line y 5 - 3x. Then identify the x-values where the line is greater than -1 and less than or equal to 20.

Verification and Solution Range

The solution to the inequality -1 ≤ 5 - 3x ≤ 20 is x ≤ -5 and x ≥ 2. Therefore, the solution in interval notation is:

[-5, 2]

This means that any value of x between -5 and 2 (inclusive) satisfies the inequality.

Conclusion

Solving inequalities is a valuable skill in algebra, applicable in various fields such as physics, engineering, and economics. By following the steps outlined above, you can systematically approach and solve complex inequalities. Whether through algebraic manipulation or graphical representation, understanding the process helps in grasping the underlying mathematical concepts.