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 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

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

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

The solution to the inequality 40 – 15(n – 1) < 1 is: ✅ n > 3.6

---

What Does This Mean?

This inequality describes a condition under which the expression 40 – 15(n – 1) is less than 1. For instance:

• If n represents the number of units produced, time elapsed, or performance inputs, this result tells us n must exceed 3.6 to satisfy the condition.
• In business contexts, it might represent a scenario where reducing inputs below 3.6 results in suboptimal output—making values beyond 3.6 more efficient.

---

Why This Breakdown Matters

Mastering such inequalities builds critical problem-solving skills:

• It enhances algebraic manipulation accuracy.
• It strengthens logical reasoning by ensuring each step follows mathematical rules.
• It prepares students and professionals for real-world modeling where variables interact linearly.

---

Summary

| Step | Operation | Result | |------|-----------|--------| | Original | 40 – 15(n – 1) < 1 | Set expanded: 55 – 15n < 1 | | Combine | — | 55 – 15n < 1 | | Subtract 55 | — | –15n < –54 | | Divide by –15 | Inequality flip | n > 54/15 = 3.6 |

Final Answer: n > 3.6

---

Understanding inequalities like this one doesn't just improve your math fluency—it helps you tackle complex problems with confidence and precision. Whether optimizing systems or evaluating thresholds, knowing how to manipulate and interpret inequalities is an essential skill in STEM, economics, data science, and beyond.

---

Keywords for SEO optimization: set 40 – 15(n – 1) < 1, solve linear inequality, algebra step-by-step, how to solve n > value, inequality solving technique, real-world inequality example, mathematical reasoning, linear expression simplification, solve for n, math tutorial inequality.