Problem 48 - Algebra 2

Question

Write \(\left(\frac{2 s}{2 s+1}-1\right) \div\left(1+\frac{2 s}{1-2 s}\right)\) in simplest form.

Step-by-Step Solution

Answer

\( \frac{-1 + 2s}{2s+1} \)

Step 1: Simplify the First Fraction

Start with the first part of the expression: \( \frac{2s}{2s+1} - 1 \). To subtract 1, we need a common denominator:\[ \frac{2s}{2s+1} - \frac{2s+1}{2s+1} = \frac{2s - (2s+1)}{2s+1} = \frac{2s - 2s - 1}{2s+1} = \frac{-1}{2s+1}. \]The simplified form of the first fraction is \( \frac{-1}{2s+1} \).

Step 2: Simplify the Second Fraction

Now, simplify the second part: \( 1 + \frac{2s}{1-2s} \). Find a common denominator:\[ \frac{1(1-2s)}{1-2s} + \frac{2s}{1-2s} = \frac{1 - 2s + 2s}{1-2s} = \frac{1}{1-2s}. \]The second expression becomes \( \frac{1}{1-2s} \).

Step 3: Divide the Simplified Fractions

You now have two fractions: \( \frac{-1}{2s+1} \) divided by \( \frac{1}{1-2s} \). To divide by a fraction, multiply by its reciprocal:\[ \frac{-1}{2s+1} \times \frac{1-2s}{1} = \frac{-1(1-2s)}{2s+1}. \]This simplifies to:\[ \frac{-1 + 2s}{2s+1}. \]

Step 4: Final Simplification (If needed)

Inspect the final expression \( \frac{-1 + 2s}{2s+1} \). There are no further simplifications available since the numerator \(-1 + 2s\) and the denominator \(2s+1\) have no common factors. Therefore, the expression is in its simplest form.

Key Concepts

Fraction Simplification

Fraction simplification makes dealing with rational expressions easier and more efficient. The goal of simplifying fractions is to make them as easy to work with as possible.
To simplify a fraction, you need to:

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by that GCF, effectively reducing the fraction.

In some problems, like in rational expressions, the process can be more complex because it may involve algebraic expressions. Take, for instance, the expression \( \frac{2s}{2s+1} - 1 \). To simplify, you first find a common denominator, allowing you to combine the terms effectively.
By achieving this, fractions become easier to work with, especially when further operations like division or addition are required.

Dividing Fractions

Dividing fractions may initially seem daunting, but it's straightforward once you understand the concept of reciprocals. To divide by a fraction, you multiply by its reciprocal. This is because dividing by a number is the same as multiplying by its multiplicative inverse.
Let's break it down:

• First, identify the two fractions involved.
• Flip the second fraction (it's now the reciprocal).
• Multiply the first fraction by this reciprocal.

For example, given \( \frac{-1}{2s+1} \div \frac{1}{1-2s} \), you convert it to \( \frac{-1}{2s+1} \times \frac{1-2s}{1} \). This approach simplifies the division process, converting it into a multiplication problem that is easier to handle.
Remember, practicing this method helps with understanding, and soon you will divide fractions with confidence!

Algebraic Expressions

Algebraic expressions like those found in rational functions include variables, constants, and operations. They represent real numbers through calculation and variable manipulation.
Understanding these expressions aids in recognizing relationships between quantities:

• Variables (e.g., \(s\) in our example) represent numbers and enable the expression to change and adapt.
• Constants (e.g., 1, -1 in the expressions) remain unchanged during problem-solving.
• Operations (addition, subtraction, multiplication, division) showcase how these quantities interact.

Consider \( \frac{2s}{2s+1}-1 \) and \( 1 + \frac{2s}{1-2s} \) from the example. Here, adjusting the fractions involves observing how the algebraic terms relate, and practicing these steps reinforces comprehension.
Algebraic expressions can seem tricky, but seeing them as grouped elements interacting helps in solving larger mathematical problems efficiently.