Solve the equation: \$ (x + 1)/(x - 3) - (3x - 2)/(x + 2) = 2\$.

The given equation

\$ (x + 1)/(x - 3) - (3x - 2)/(x + 2) = 2\$

To solve this equation, let’s find the least common denominator (LCD) of the rational expressions, which is \$(x - 3)(x + 2)\$:

\$ ((x + 1)(x + 2) - (3x - 2)(x - 3)) / ((x - 3)(x + 2)) = 2 \$

Expanding and simplifying the numerator:

\$ ((x^2 + 3x + 2) - (3x^2 - 11x + 6)) / ((x - 3)(x + 2)) = 2 \$

\$ (-2x^2 + 14x - 4) / ((x - 3)(x + 2)) = 2/1 \$

By using cross multiplication, and then simplified

\$ -2x^2 + 14x - 4 = 2(x - 3)(x + 2) \$

\$ -2x^2 + 14x - 4 = 2(x^2 - x - 6) \$

\$ -2x^2 + 14x - 4 = 2x^2 - 2x - 12 \$

Bringing all terms to one side,then simplified

\$ 2x^2 - 2x - 12 + 2x^2 - 14x + 4 = 0 \$

\$ x^2 - 4x - 2 = 0 \$

Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula

\$ x = (-b ± sqrt(b^2 - 4ac)) / (2a) \$

In this case, a = 1, b = - 4, and c = - 2. Plugging these values into the quadratic formula, we have:

\$ x = (-(-4) ± sqrt((-4)^2 - 4 * 1 * (-2))) / (2 * 1) \$

\$ x = (4 ± 2sqrt(6)) / 2 \$

\$ x = 2 ± sqrt(6)\$

Therefore, the solutions to the original equation are \$ x = 2 + sqrt(6) and x = 2 - sqrt(6) \$.

These solutions don’t match the ones we found, which were x = \$ 2 + \sqrt6,2 - \sqrt6\$. Therefore, this option is incorrect

\$ 2 - \sqrt6,-2 - \sqrt6\$

This option proposes one correct solution, x = \$ 2 + \sqrt6\$, but pairs it with x= \$2 - \sqrt6\$. which doesn’t match the solution derived from our calculation. Therefore, this option is incorrect

\$ 2 + \sqrt6,-2 - \sqrt6\$

Like option B, it is incorrect because it only partially matches the correct set of solutions

\$ -2 + \sqrt6,2 - \sqrt6\$

Option D suggests both solutions involve 2 as their base value, but with different signs for the square root term. Therefore, this option is incorrect

\$ 2 + \sqrt6,2 - \sqrt6\$

Option

D

Can you briefly tell me what you’ve learned and understood in today’s lesson?