Question

solve for w:
\frac{1}{(w + 2)(w + 1)} = \frac{3}{w + 1} + 2
select all that apply.
\square w = -\frac{3}{2}
\square w = -3
\square no solution
\square w = \frac{3}{2}
\square w = 4
\square w = 3
\square w = -4
\square w = 0

question 37
2.5 pts
pam can paint a fence in 5 hours, and tom can paint the same fence in 8 hours. how long does it take them to paint the fence if they work together? round to the nearest tenth.
_(blank)_ hours
type your numerical answer below.

Explanation:

First Problem (Solve for $w$)

Step1: Eliminate denominators

Multiply all terms by $(w+2)(w+1)$ (note $w
eq -1, -2$ to avoid division by zero):
$$1 = 3(w+2) + 2(w+2)(w+1)$$

Step2: Expand all terms

$$1 = 3w + 6 + 2(w^2 + 3w + 2)$$
$$1 = 3w + 6 + 2w^2 + 6w + 4$$

Step3: Simplify to quadratic form

Combine like terms and rearrange:
$$2w^2 + 9w + 9 = 0$$

Step4: Factor the quadratic

$$(2w + 3)(w + 3) = 0$$

Step5: Solve for $w$ and check validity

Set each factor to zero:
$2w + 3 = 0 \implies w = -\frac{3}{2}$; $w + 3 = 0 \implies w = -3$
Neither value makes original denominators zero, so both are valid.

Step1: Define work rates

Pam's rate: $\frac{1}{5}$ fence per hour; Tom's rate: $\frac{1}{8}$ fence per hour.

Step2: Combined work rate

Add individual rates:
$$\text{Combined rate} = \frac{1}{5} + \frac{1}{8} = \frac{8 + 5}{40} = \frac{13}{40}$$

Step3: Calculate time for 1 fence

Time = $\frac{1}{\text{Combined rate}}$:
$$\text{Time} = \frac{40}{13} \approx 3.1$$

Answer:

$w=-\frac{3}{2}$, $w=-3$

---

Second Problem (Painting Fence)