Problem 18 - Algebra 1

Question

Simplify the expression. $$\frac{5-2 x}{-2} \cdot \frac{24}{10-4 x}$$

Step-by-Step Solution

Answer

The simplified expression is 16

Step 1: Simplify the Expression

Simplify the expression in the numerator of the second fraction to give \(\frac{5-2x}{-2} \cdot \frac{24}{6(2 - x)}\)

Step 2: Factorize the First Fraction

Factorize the numerator of the first fraction to get \(-1 \cdot \frac{ (2x - 5)}{2} \cdot \frac{24}{6(2 - x)}\)

Step 3: Reverse the Second Numerator

Factorize the denominator of the second fraction and swap the order to align the like terms on the top and bottom \(-1 \cdot \frac{2x - 5}{2} \cdot \frac{24}{-6(x - 2)}\)

Step 4: Simplification

Simplify \((2x - 5)\) and \((x - 2)\) to give: \(-1 \cdot \frac{-4x + 10}{2} \cdot \frac{24}{-6}\)

Step 5: Further Simplification

Simplify by cancelling out terms to give: \(-1 \cdot -4 \cdot 4 = 16 \)

Key Concepts

Fraction Operations

Fraction operations involve the processes of adding, subtracting, multiplying, and dividing fractions. When working with fractions, the primary goal is often to simplify the expression as much as possible, making it easier to understand and work with. In this exercise, we are asked to multiply two fractions: \( \frac{5-2x}{-2} \) and \( \frac{24}{10-4x} \). The multiplication of fractions involves multiplying the numerators together and the denominators together.

• Multiplying Numerators and Denominators: To multiply fractions, simply multiply the numerators (top parts) across and the denominators (bottom parts) across as well. For instance, you're looking to perform: \((5-2x) \times 24\) for the numerator and \(-2 \times (10 - 4x)\) for the denominator.
• Simplification After Multiplication: Once you multiply, it's crucial to look for opportunities to simplify the products. This can involve canceling common factors between the numerator and denominator, which often makes the fraction easier to work with or interpret.

Understanding these basics can help make confusing expressions look simpler, and consequently, make calculations or further transformations much easier.

Factoring Algebraic Expressions

Factoring algebraic expressions is a key step in simplifying complex algebraic fractions. It involves breaking down expressions into simpler components that are the product of factors. In the given exercise, we need to factorize the given numerators and denominators to aid in the simplification of our expression.

• Identify and Apply Factors: Begin by identifying common terms or patterns that enable you to factor the expression. For example, the expression \(5-2x\) can be rewritten in terms of \(-1(2x-5)\), simplifying the expression and making the roots or solutions more visible.
• Using Factoring to Simplify: Once factored, the new expressions should highlight common elements. For instance, when dealing with \(24\) in the numerator of the second fraction, notice that \(10-4x\) can be recast as \(6(2-x)\) by factoring out 2. This helps reveal how parts of the expression might cancel each other out.

Using factoring allows for the cancellation of terms when multiplying fractions, leading to simpler and more manageable expressions.

Numerator and Denominator Simplification

Simplifying the numerator and denominator is a crucial step in dealing with fraction operations more efficiently. This involves reducing expressions to their simplest form and often involves canceling out matching terms in both the numerator and denominator.

• Cancellation of Common Factors: After factoring, the fraction \(\frac{(2x-5)(x-2)}{-2 \times -6}\) can have common elements divided out. This makes terms disappear, simplifying calculations. For instance, \((x-2)\) appears both in the numerator and denominator, enabling cancellation.
• Recognition and Removal of Unnecessary Signs: Negative signs can complicate expressions unnecessarily. Notice how turning \(10-4x\) into its negative factor form \(-6(x-2)\) inverts and aligns the expression for improved clarity. Such changes can make the rest of the simplification process straightforward.

Correct and thorough simplification not only makes the expression cleaner but also can reveal insights into the properties of algebraic expressions through this clearer form.