Set 40 − 15(n−1) < 1 → 40 − 15n + 15 < 1 → 55 − 15n < 1 → −15n < −54 → n > 54/15 = <<54/15=3.6>>3.6 - Veritas Home Health
Understanding the Inequality: Set 40 – 15(n – 1) < 1 Step-by-Step Breakdown
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
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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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
