Question

Which expression is equivalent to (x + 3)/(x² - 2x - 3) ÷ (x²+2x - 3)/(x+1) if no denominator equals zero?

a. 1/(x² -2x-3)
b. 1/(x² -4x+3)
c. 1/(x² +2x-3)
d. (x+3)/(x+1)

Answer 1

To find the equivalent expression, divide the fractions by multiplying by the reciprocal and then simplifying. None of the provided options exactly matches the result when the original expression is fully simplified.

Step-by-step explanation:

To find the equivalent expression for (x + 3)/(x² - 2x - 3) ÷ (x² + 2x - 3)/(x+1), we need to perform division of fractions, which is the same as multiplying by the reciprocal.

First, factor the denominators:

• x² - 2x - 3 = (x - 3)(x + 1)
• x² + 2x - 3 = (x + 3)(x - 1)

Now we write the division as a multiplication with the reciprocal:

((x + 3) / ((x - 3)(x + 1))) × ((x + 1) / ((x + 3)(x - 1)))

Next, simplify by canceling out the common terms:

((x + 3) / (x - 3)(x + 1)) × (1 / (x - 1)) = (x + 3) / (x - 3)(x - 1)

We cannot cancel out any further terms, so the equivalent expression is:

(x + 3) / (x - 3)(x - 1), which doesn't match exactly to any of the given options a, b, c, or d, suggesting a possible typo or mistake in the provided options, or in the expression given by the student.