Solve the following equation: $\frac{2}{(2r - 1)}$ - $\frac{5}{3}$ =  $\frac{1}{(r + 2)}$

1. Analyze the question

We are given a rational equation:

Our goal is to find the values of $r$ that satisfy this equation.

2. Explain the concept

To solve an equation involving fractions, we need to eliminate the denominators by multiplying every term by the Least Common Multiple (LCM) of the denominators. Once the fractions are cleared, we will have a quadratic equation, which we can solve by factorization or the quadratic formula.

3. Show the solution

Step 1: Identify the LCM The denominators are $(2r - 1)$, $3$, and $(r + 2)$. The LCM is $3(2r - 1)(r + 2)$.

Step 2: Multiply every term by the LCM

Step 3: Simplify

Expand the brackets:

Step 4: Rearrange into standard quadratic form ($ar^2 + br + c = 0$)

Divide the entire equation by $-5$ to simplify:

Step 5: Factorize We need two numbers that multiply to $(2 \times -5) = -10$ and add to $3$. These are $5$ and $-2$.

Setting each factor to zero: $2r + 5 = 0 \implies r = -\frac{5}{2}$ $r - 1 = 0 \implies r = 1$

The solutions are $1$ and $-\frac{5}{2}$.

4. Identify the answer

Comparing our results ($1, -\frac{5}{2}$) with the given options, we see that it matches option B.

5. Tip

When dealing with rational equations, always check your final answers against the original equation. If any value of $r$ makes a denominator zero (e.g., $r = 0.5$ or $r = -2$ in this case), that value is "undefined" and must be rejected. Here, both our answers are valid!