Problem 37 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 7 x 4 x 2-9 x+2-2 x-2 $$

Step-by-Step Solution

Answer

The simplified expression is \( 56 - 11x \).

Step 1: Expand the Expression

The expression given is \( 7 \times 4 \times 2 - 9x + 2 - 2x - 2 \). Start by expanding the numerical multiplication: \( 7 \times 4 \times 2 = 56 \). So the expression becomes \( 56 - 9x + 2 - 2x - 2 \).

Step 2: Simplify the Numerical Terms

Identify the numerical terms in the expression: 56, +2, and -2. Now combine these numbers together: \( 56 + 2 - 2 = 56 \).

Step 3: Simplify the Variable Terms

Now, look at the terms involving the variable 'x': -9x and -2x. Combine these terms by adding them together: \(-9x - 2x = -11x \).

Step 4: Combine Simplified Terms

Now that both the numerical and the variable parts have been simplified, the expression can be written in a simpler form. Combine the results from Step 2 and Step 3 to get the final simplified expression: \( 56 - 11x \).

Key Concepts

Expression Expansion

Expression expansion involves breaking down or "expanding" the parts of an expression to make it easier to simplify. In the given exercise, the expression starts with a multiplication of numbers: \(7 \times 4 \times 2\). Instead of tackling this as a whole, we perform the arithmetic operation first. This simplifies to \(56\).

Breaking down expressions into smaller parts is a strategy that can help make complex expressions more manageable. It is especially useful when multiple operations are involved.

Combining Like Terms

Combining like terms helps in simplifying expressions by grouping similar components together. In this context, 'like terms' are those that share the same variable (or are all numeric). Let's take the expression \(56 - 9x + 2 - 2x - 2\).

For numerical values:

• Identify: 56, +2, and -2.
• Combine: \(56 + 2 - 2 = 56\).

For variable terms:

• Identify: -9x and -2x.
• Combine: \(-9x - 2x = -11x\).

This simplification reduces the complexity and makes the calculation straightforward.

Mathematical Operations

Mathematical operations like addition, subtraction, multiplication, and division form the backbone of simplifying expressions. To work through the expression given, we start with multiplication \(7 \times 4 \times 2\), which simplifies to \(56\).

Afterward, we address addition and subtraction among the numerical terms and variable terms separately. Each operation needs careful execution to maintain the integrity of the expression. This ordered method ensures that each step builds correctly towards a simpler form.

Variable Simplification

Variable simplification focuses on making expressions involving variables like 'x' as concise as possible. In the problem, there are two variable terms: \(-9x\) and \(-2x\).

By combining them, we have:

• Write: \(-9x - 2x\).
• Simplify: \(-11x\).

This process makes it easier to see the actual quantity or coefficient linked to 'x'. Simplifying variable expressions can be particularly useful for solving equations or for further mathematical analysis.