Simplify the expression:
$-\frac{3}{2}(4x - 2)$
The problem requires simplifying the expression \(-\frac{3}{2}(4x - 2)\). This is a linear algebraic expression where we need to apply the distributive property, which states that for any real numbers \(a, b,\) and \(c\), \(a(b - c) = ab - ac\). In our specific case, the constant factor outside the parentheses is \(-\frac{3}{2}\), and the terms inside are \(4x\) and \(2\).
To simplify the expression, we distribute the multiplier \(-\frac{3}{2}\) to each term inside the parentheses:
\[-\frac{3}{2}(4x - 2) = \left(-\frac{3}{2} \cdot 4x\right) - \left(-\frac{3}{2} \cdot 2\right).\]
First, we calculate the product of \(-\frac{3}{2}\) and \(4x\):
\[-\frac{3}{2} \cdot 4x = -3 \cdot \left(\frac{4}{2}\right)x = -3 \cdot 2x = -6x.\]
Next, we calculate the product of \(-\frac{3}{2}\) and \(-2\):
\[-\frac{3}{2} \cdot (-2) = 3.\]
Combining these, we get \(-6x + 3\).
The expression \(-\frac{3}{2}(4x - 2)\) simplifies to \(-6x + 3\). This simplified form represents the same linear relationship as the original expression, but in a more compact and standard form. Thus, the final simplified expression is \(\boxed{-6x + 3}\).
In many practical scenarios, such as budget management or inventory analysis, you might have formulas that account for deductions and adjustments. For example, if you have a budget deduction factor that multiplies a series of variable and fixed costs, say \(-1.5 \times (4x - 2)\), you would use this algebraic distribution to quickly calculate your net adjustment. Simplifying such expressions helps in performing faster calculations and gaining clearer insight into how changing the variable cost \(x\) impacts the total.