Completely simplify $\frac{\frac{x^2+1}{3}}{\frac{x+1}{3}}$ and state any restrictions on the variable.
A. $\frac{x^2+1}{x+1}, x \neq-1$
B. $\frac{3\left(x^2+1\right)}{x+1}, x \neq-1$
C. $x-1, x \neq-1$
D. $\frac{x^2+1}{3(x+1)}, x \neq-1$
Answer (2)
The simplified expression of the complex fraction is x + 1 x 2 + 1 with the restriction that x = − 1 .
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Simplify the complex fraction by multiplying the numerator and denominator by 3: 3 x + 1 3 x 2 + 1 = x + 1 x 2 + 1 .
Determine the restriction on x by ensuring the denominator is not zero: x + 1 = 0 .
Solve for x to find the restriction: x = − 1 .
State the simplified expression with the restriction: x + 1 x 2 + 1 , x = − 1 .
Explanation
Understanding the Problem We are given a complex fraction to simplify: 3 x + 1 3 x 2 + 1 . We also need to identify any restrictions on the variable x .
Simplifying the Fraction To simplify the complex fraction, we can multiply both the numerator and the denominator by 3: 3 x + 1 3 x 2 + 1 = 3 x + 1 × 3 3 x 2 + 1 × 3 = x + 1 x 2 + 1
Finding Restrictions Now, we need to find the restrictions on x . The denominator of the simplified fraction cannot be zero, so we must have x + 1 = 0 . Solving for x , we get x = − 1 .
Final Answer Therefore, the simplified expression is x + 1 x 2 + 1 , with the restriction that x = − 1 .
Examples
Imagine you're baking a cake and need to adjust the recipe. This problem is like simplifying the ratio of ingredients to make the recipe easier to follow. Simplifying complex fractions helps in many real-world situations, such as adjusting ratios in recipes, calculating rates in physics, or simplifying formulas in engineering. By understanding how to simplify these fractions, you can make complex calculations more manageable and avoid errors.
