Question

Solve for \(\frac{12\left(\frac{5 x-2 x}{2}\right)}{6}\) (A) 24 (B) \(\frac{1}{4}\) (C) \(\frac{1}{4}\) (D) 6

Short answer

The expression simplifies to \(3x\), which doesn't match any options given.

Step-by-step solution

Understanding the Inner Expression

First, focus on simplifying the expression \( \frac{5x - 2x}{2} \). Start by subtracting the two terms in the numerator.

Simplifying the Numerator

Calculate \(5x - 2x\), which results in \(3x\). Now your inner expression becomes \(\frac{3x}{2}\).

Multiplying by 12

Next, take the simplified expression \(\frac{3x}{2}\) and multiply it by 12, as per the outer function: \(12 \times \frac{3x}{2}\).

Simplifying the Multiplication Step

Perform the multiplication: \(12 \times \frac{3x}{2} = \frac{36x}{2}\).

Simplifying the Division

Now, simplify the fraction by dividing the numerator by the denominator: \(\frac{36x}{2} = 18x\).

Dividing by 6

Finally, divide the result by 6: \(\frac{18x}{6} = 3x\).

Conclusion

The simplified expression evaluates to \(3x\), which does not depend on specific values of \(x\). Since none of the options match \(3x\), reassess the interpretations.

Key concepts

Numerator and Denominator

When dealing with fractions, understanding numerators and denominators is crucial. A fraction consists of two parts: the numerator, which sits on top, and the denominator, which is below the line. The numerator represents how many parts of a whole we have, while the denominator tells us into how many equal parts the whole is divided.

In the expression \( \frac{5x - 2x}{2} \), \(5x - 2x\) is the numerator. We first simplify this by performing the subtraction within it, yielding \(3x\). Following that, \(2\) serves as the denominator, determining how the result is divided. Understanding these components helps simplify and solve complex expressions.

• Numerator: Represents parts of a whole.
• Denominator: Indicates the division of the whole.

By simplifying this fraction step-by-step, you handle complex expressions effectively, starting with simplifying the numerator before addressing the fraction as a whole.

Math Problem Solving

Math problem solving involves breaking down complex problems into simpler steps and systematically working through them. It's crucial to read the problem carefully to identify the operations involved and the sequence in which they should be conducted.
For instance, when solving \(\frac{12\left( \frac{3x}{2} \right)}{6}\), clear steps are necessary:

• Simplify the inner expressions first, as done with \(5x - 2x\) in the numerator.
• Follow the order of operations — start by simplifying multiplications and divisions before handling subtractions, if present.
• Once simplified, reevaluate the answer to ensure logical consistency.

By approaching problems methodically, you effectively navigate complex algebraic expressions and find the simplest form possible. This structured method also helps avoid errors, particularly with fractions and signed numbers.

Expression Evaluation

Evaluating expressions involves substituting numbers or simplifying expressions according to algebraic rules. In the given problem, evaluating the expression involves several stages, each needing careful arithmetic operations to deduce the simplest form.
Specifically, you first reduced \(5x - 2x\) to \(3x\). Following this, multiplying \(3x\) by 12 was needed, converting it into \(\frac{36x}{2}\). This expression was then simplified to produce \(18x\). Lastly, dividing \(18x\) by 6 yielded \(3x\).
Key steps in evaluating expressions include:

• Simplification: Streamline each component by performing arithmetic operations.
• Substitution: Replace variables with numbers where applicable.
• Reassessment: Continually check for errors or misinterpretations.

This process ensures your final result accurately reflects the operations and logical structure of your original expression, providing a clearer understanding of how algebraic expressions operate.