Problem 67 - College Algebra

Question

Simplify each complex rational expression. $$\frac{\frac{3}{x-2}-\frac{4}{x+2}}{\frac{7}{x^{2}-4}}$$

Step-by-Step Solution

Answer

The simplified form of the given complex rational expression is \(x = 7\).

Step 1: Identify the least common denominator

First, find the least common denominator (LCD) of the complex expression, which is the product of the denominators of each rational expression involved in the problem. Here, the denominators are \(x-2\), \(x+2\), and \(x^{2}-4\). The term \(x^{2}-4\) can be factored into \((x-2)(x+2)\), so the LCD is \((x-2)(x+2)\).

Step 2: Rewrite each term

Next, rewrite each term in the problem by multiplying the term by the LCD over itself. This effectively converts each term into an easier to deal with single fraction. Do this for all terms to get: \\(x-2)(x+2)\left(\frac{3}{x-2}\right)-(x-2)(x+2)\left(\frac{4}{x+2}\right)=(x-2)(x+2)\left(\frac{7}{x^{2}-4}\right)\\

Step 3: Simplify the fractions

Now, simplify the fractions. Terms will cancel from the numerator and denominator that are same: \\(3(x+2)-4(x-2)=7\\

Step 4: Further simplify the equation

Once again, simplify the equation by performing the distribution: \\3x+6 - 4x+8= 7 \\

Step 5: Combine like terms

Combine like terms on left side, this results in \\-x +14 = 7\\

Step 6: Solve for x

Solve for x by subtracting 14 from both sides of the equation to isolate x, yielding \\-x = -7. Multiply through by -1 to get x = 7 as the final result.

Step 7: Final simplified expression

The final simplified expression can inserted back in the original problem to verify. If it checks out, then the simplification is correct.