Problem 4 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$\frac{x^{2}-1}{x} \cdot \frac{2 x}{3 x-3}$$

Step-by-Step Solution

Answer

The simplification of the given expression results in \(\frac{2x+2}{3}\).

Step 1: Factor Numerator in First Fraction

The numerator \(x^{2}-1\) can be factored using the difference of squares rule to get: \(x^{2}-1= (x+1)(x-1)\), thus resulting in the fraction \(\frac{(x+1)(x-1)}{x}\).

Step 2: Factor Denominator in Second Fraction

Note that the denominator in the second fraction, \(3x - 3\), can be factored by taking out a common factor of 3: \(3x-3=3(x-1)\), hence the whole fraction becomes \(\frac{2x}{3(x-1)}\).

Step 3: Rewrite Given Expression

Now, rewrite the whole expression with the factored terms: \(\frac{(x+1)(x-1)}{x} \cdot \frac{2x}{3(x-1)}\).

Step 4: Cancel Similar Terms

Next, cancel out similar terms from the numerator and denominator. Notice that x from each fraction can be cancelled, and also (x - 1): After simplification we obtain: \(\frac{(x+1) \cdot 2}{3}\).

Step 5: Final Simplification

Finally, multiply through the numerator to obtain: \(\frac{2x+2}{3}\). This is the simplified form of the given expression.