Problem 58 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{2 x}{x-2} \div \frac{x+2}{x} \div \frac{7 x}{x^{2}-4} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2}{7} \).

Step 1: Understand the Problem

We need to simplify the expression \( \frac{2x}{x-2} \div \frac{x+2}{x} \div \frac{7x}{x^2-4} \). The expression involves division of fractions, which we can handle by multiplying by the reciprocal.

Step 2: Identify the Reciprocals

The expression \( \frac{a}{b} \div \frac{c}{d} \) can be rewritten as \( \frac{a}{b} \times \frac{d}{c} \). Similarly, we identify the reciprocals of the fractions involved: \( \frac{x}{x+2} \) and \( \frac{x^2-4}{7x} \).

Step 3: Factorize the Denominator

Recognize that \( x^2-4 \) is a difference of squares, which can be factorized as \( (x-2)(x+2) \). We rewrite the expression as:\[ \frac{2x}{x-2} \times \frac{x}{x+2} \times \frac{(x-2)(x+2)}{7x} \]

Step 4: Multiply the Fractions

Now, multiply the numerators and denominators:- Numerator: \( 2x \times x \times (x-2)(x+2) = 2x(x-2)(x+2) \)- Denominator: \( (x-2) \times (x+2) \times 7x = 7x(x-2)(x+2) \)This gives:\[ \frac{2x(x-2)(x+2)}{7x(x-2)(x+2)} \]

Step 5: Simplify the Expression

Cancel the common factors in the numerator and denominator. The terms \((x-2)(x+2)\) and \(x\) cancel out:\[ \frac{2}{7} \]Thus, the simplified expression is \( \frac{2}{7} \).

Key Concepts

Division of Fractions

Division with fractions might seem tricky at first, but it becomes straightforward when you know the correct method. Dividing by a fraction is the same as multiplying by its reciprocal. This means you flip the numerator and denominator of the fraction you are dividing by. Here's a step-by-step guide to do it:

• Identify the fractions involved in the division.
• Invert (flip) the fraction you are dividing by to find its reciprocal.
• Change the division sign to multiplication.
• Multiply the fractions as usual by multiplying the numerators and then the denominators.

For example, in the expression \( \frac{2x}{x-2} \div \frac{x+2}{x} \), we find the reciprocal of \( \frac{x+2}{x} \) as \( \frac{x}{x+2} \) and multiply: \[ \frac{2x}{x-2} \times \frac{x}{x+2} \]This technique brings a lot of ease to dealing with division of fractions.

Difference of Squares

The difference of squares is a special factorization identity that applies when you have two perfect squares subtracted from each other. It can be represented as:

• \( a^2 - b^2 = (a - b)(a + b) \)

This identity is very handy and pops up quite frequently in algebraic simplifications. In our problem, we deal with \( x^2 - 4 \). Notice that both are perfect squares: \( x^2 \) is a square of \( x \) and 4 is a square of 2. Using the difference of squares identity, \( x^2 - 4 \) factorizes to: \[ (x-2)(x+2) \]Recognizing and applying this property makes simplification easier, as it divides the expression into simpler, manageable parts that can lead to cancellation of terms.

Multiplying Fractions

Once division of fractions is handled through multiplication by the reciprocal, we move on to multiplying the fractions. The rule is simple:

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

In the example given in the exercise:\[ \frac{2x}{x-2} \times \frac{x}{x+2} \times \frac{(x-2)(x+2)}{7x} \]We multiply the numbers across:

• Numerator: \( 2x \times x \times (x-2)(x+2) = 2x(x-2)(x+2) \)
• Denominator: \( (x-2) \times (x+2) \times 7x = 7x(x-2)(x+2) \)

After multiplication, check for any common factors that can help in simplifying the expression further. This systematic approach will always lead you to the simplest form of the fractions.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler 'factor' polynomials that multiply to the original one. It’s a crucial skill in algebra, essential for simplifying expressions and solving equations.
In our case, we initially find the polynomial \( x^2 - 4 \), which is factored using the difference of squares formula into \( (x-2)(x+2) \). Factoring helps us:

• Identify shared factors in expressions, aiding in the simplification process.
• Convert complex algebraic terms into easier, linear factors.

Besides the difference of squares, remember other forms of factoring such as:

• Factoring out a common factor.
• Trinomial factoring.
• Grouping.

Recognizing when and how to apply these different techniques will simplify even the most daunting algebraic expressions.