Problem 49 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ 2 x-\\{3 x-[x-(2 x-1)]\\} $$

Step-by-Step Solution

Answer

The simplified expression is \(-2x+1\).

Step 1: Remove parentheses.

In order to begin to simplify the expression, we'll first remove the parentheses and distribute the negative signs. Given expression: \[ 2x -\{3x -[x-(2x-1)]\} \] Let's distribute the negative sign inside the inner brackets \( [~]\): \[ 2x -\{3x - (x - 2x +1)\} \] Now remove the parentheses: \[ 2x- \{3x- x+ 2x-1\} \]

Step 2: Combine like terms inside the brackets.

Combine the similar terms (the terms containing 'x') inside the bracket: \[ 2x- \{3x-1x+2x-1\} \] Add or subtract the terms with 'x': \[ 2x- \{4x-1\} \]

Step 3: Distribute the negative sign.

Distribute the negative sign outside the bracket to each term inside the bracket: \[ 2x - 4x +1 \]

Step 4: Combine like terms.

Combine the similar terms (the terms containing 'x'): \[ 2x-4x = -2x \] Hence, the simplified expression is: \[ -2x+1 \]

Key Concepts

Simplifying Expressions

Simplifying algebraic expressions makes them easier to understand and solve. It involves reducing complex expressions to simpler forms by eliminating unnecessary parts. When you simplify, you typically work with parentheses or other grouping symbols.
First, handle what's inside the parentheses or brackets. Make sure to work from the innermost parts outwards. This approach prevents mistakes and accurately changes the expression's structure. While simplifying, always keep track of all terms and signs.
A simplified expression generally has fewer terms. Simplifying also reveals the core components, which makes further algebraic operations much easier to perform.

Combining Like Terms

Like terms in an expression have the same variable raised to the same power, such as '3x' and '5x'. Combining like terms means adding or subtracting these to consolidate the expression.

• First, identify all like terms in the expression.
• Group these terms together for clarity.
• Add or subtract numerical coefficients to combine them.

For example, in the expression \(2x - 3x + 4 \), \(2x\) and \(-3x\) are like terms. When combined, they form \(-x\). This process helps in simplifying expressions by reducing the number of terms you need to handle.

Distributive Property

The distributive property in algebra allows you to multiply a single term by each term inside a set of parentheses. It is represented by the rule:
\(a(b + c) = ab + ac\).
This property is crucial for simplifying expressions, especially when dealing with parentheses and multiple terms.

• Apply the multiplication to each term inside the brackets.
• Rewrite the expression without parentheses, ensuring you multiply correctly.

For instance, if you have \(2(x - 3)\), using the distributive property gives you \(2x - 6\). Always watch out for signs, especially negative ones, when applying this property. It can change the entire expression if not handled properly.

Negative Signs in Algebra

Negative signs can be tricky in algebra, particularly when combined with parentheses. Proper handling of negative signs is essential to avoid errors in simplifying and solving expressions.

• Remember that a negative sign outside a bracket changes the sign of each term within when distributed.
• For example, \(-(x + 2)\) becomes \(-x - 2\).
• Keep track of multiple negatives. Two negatives become positive: \(-(-x) = x\).

Getting accustomed to negative signs, especially when nested within several layers of parentheses, requires practice. But once mastered, it greatly improves your algebraic proficiency by preventing miscalculations that lead to incorrect solutions.