Understanding the Inequality: Set 40 – 15(n – 1) < 1 Step-by-Step Breakdown

Solving inequalities is a foundational skill in algebra, especially when analyzing relationships between variables—especially real-world applications like best-case scenarios, break-even analysis, or performance benchmarks. One common but often confusing form is:

> Set 40 – 15(n – 1) < 1

Understanding the Context

Understanding how to manipulate this inequality correctly unlocks deeper insight into linear relationships and helps solve problems efficiently. In this article, we’ll break down this inequality step-by-step to clarify each transformation, arrive at the correct solution, and explore its practical meaning.

Step-by-Step Explanation

Let’s dissect the original inequality:
Set 40 – 15(n – 1) < 1

Set 40 − 15(n−1) < 1 → 40 − 15n + 15 < 1 → 55 − 15n < 1 → −15n < −54 → n > 54/15 = <<54/15=3.6>>3.6

Key Insights

Step 1: Expand the parentheses

Start by distributing the −15 across the expression inside the parentheses:

> 40 – 15(n – 1) = 40 – 15n + 15
(Because –15 × –1 = +15)

Now the inequality becomes:
40 – 15n + 15 < 1

Step 2: Combine like terms

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Final Thoughts

Combine the constant terms on the left-hand side:
40 + 15 = 55
So,
55 – 15n < 1

Step 3: Isolate the term with n

Subtract 55 from both sides to move the constant to the right:
–15n < 1 – 55
–15n < –54

Step 4: Solve for n

Now divide both sides by −15. Important: When dividing or multiplying both sides of an inequality by a negative number, you must reverse the inequality sign:

> n > (–54) ÷ (–15)
n > 54/15

Now simplify the fraction:

54 ÷ 15 = 3.6
So,
n > 3.6

Final Result