Verified Answer :

  • Simplify the expression \(9n - 3 - 6 + 5n\) by grouping like terms: \(9n + 5n = 14n\) and \(-3 - 6 = -9\), giving \(14n - 9\).
  • Expand the expression \(-2(x - 6)\) using the distributive property: \(-2 \times x = -2x\) and \(-2 \times -6 = 12\), resulting in \(-2x + 12\).
  • The fragment "TRIANGLE, BOX LIKE TER" is interpreted as a non-mathematical descriptor.
  • Final simplified expressions are: \(\boxed{14n - 9}\) and \(\boxed{12 - 2x}\).

Explanation

  1. Analysis and Simplification of Expressions

The problem presents a collection of mathematical fragments. The objective is to analyze and interpret these expressions correctly.

  1. Item 1: "TRIANGLE, BOX LIKE TER"

This fragment appears to be a descriptive label or a non-standard prompt. Without additional numerical data or geometric properties, it is not a solvable mathematical problem. We interpret it as a reference to basic geometric shapes.

  1. Item 2: Algebraic Simplification

We are given the expression \(9n - 3 - 6 + 5n\).
To simplify, we group the terms involving the variable \(n\) and the constant terms:
\((9n + 5n) + (-3 - 6)\)
Calculating these sums, we get:
\(14n - 9\)
The provided intermediate step, \(3 - 6 = -3\), seems to be a misrepresentation of the operation \(-3 - 6\). The actual result of \(-3 - 6\) is \(-9\). Thus, the full simplification is \(14n - 9\).

  1. Item 3: Expression Expansion

We are given the expression \(-2(x - 6)\).
To expand this expression, we use the distributive property, multiplying \(-2\) by each term inside the parentheses:
\(-2 \times x + (-2) \times (-6)\)
\(-2x + 12\)
This can also be written as \(12 - 2x\).

Examples

In practical applications like budget management, keeping track of different cost categories is essential. If you have a base cost of \(14n\) dollars and you need to subtract various fixed fees (like \(-3\) and \(-6\)), you are performing an algebraic simplification similar to the one we just solved, resulting in \(14n - 9\). Understanding how to correctly simplify these expressions allows you to predict your total costs accurately.

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