Problem 97 - College Algebra

Question

Write each algebraic expression without parentheses. \(-(-14 x)\)

Step-by-Step Solution

Answer

The expression \(-(-14 x)\) without parentheses is \(14 x\).

Step 1: Identify the Negative Symbols

Recognize the negative symbols before the brackets. There are two negative symbols, one outside the brackets and one inside.

Step 2: Apply the Rule of Negatives

Apply the rule: Negative times negative equals positive. Therefore, \(-(-14 x)\) equals \(+14 x\).

Step 3: Final Expression

The expression can be simplified to \(14 x\) because a positive sign before an algebraic term is generally not written.

Key Concepts

Parentheses Elimination

In algebra, parentheses are often used to group terms and clarify the order of operations. Removing these parentheses is a key step in simplifying expressions. When you see parentheses around terms with operations, such as

• Addition or subtraction
• Multiplication

you must follow specific rules to eliminate them correctly.
For example, in the expression \(-(-14x)\), the parentheses indicate that the operation inside should be addressed first. Once you recognize the symbols and understand their interaction, you can rewrite the expression without parentheses. By applying the operations step by step, you will remove the parentheses while keeping the expression equivalent.

Negatives in Algebra

Negative symbols in algebra can be tricky as they affect the terms they are attached to—making their handling crucial for accurate simplification. In the expression \(-(-14x)\), two negatives appear:

The first negative is outside the parentheses.

The second negative is inside the parentheses with the term \(14x\).

When dealing with negatives:

• A negative multiplied by another negative becomes positive.
• This rule helps simplify expressions involving multiple negatives.

So, applying this rule to our example, \(-(-14x)\) becomes \(+14x\),which is the same as just writing \(14x\). This simplification reduces the confusion that negatives might introduce, resulting in a cleaner and more straightforward expression.

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form while maintaining equality. It's about making an expression easier to understand and work with.
The process essentially includes:

• Removing unnecessary signs, such as eliminating extra positives
• Combining like terms when possible

In our example of \(14x\),

• The term is already simplified because it is a direct result of the expression \(-(-14x)\) after applying sign rules.
• There's no need to include the positive sign in front of \(14x\)

This practice not only makes the expression more aesthetically pleasing but also prepares it for further operations if necessary, ensuring that you always have a clear and concise form.