Question

name neysindee velez

1. \\(\dfrac{\dfrac{u}{2} - \dfrac{u}{u - 4}}{\dfrac{4}{u - 4} - \dfrac{25}{2}}\\)
2. \\(\dfrac{\dfrac{1}{5} + \dfrac{1}{3}}{\dfrac{25}{3} + \dfrac{a^2}{a + 1}}\\)
3. \\(\dfrac{\dfrac{x}{9} + \dfrac{4}{5}}{\dfrac{2}{3} + \dfrac{x}{5}}\\)
4. \\(\dfrac{\dfrac{9}{m} - \dfrac{3}{m + 3}}{\dfrac{m - 1}{3} + \dfrac{m}{m + 3}}\\)

Answer:

Let's solve problem 17 step by step. The problem is a complex fraction: \(\frac{\frac{u}{2}-\frac{u}{u - 4}}{\frac{4}{u - 4}-\frac{25}{2}}\)

Step 1: Simplify the numerator \(\frac{u}{2}-\frac{u}{u - 4}\)

To subtract these two fractions, we need a common denominator, which is \(2(u - 4)\).
\[

$$\begin{align*} \frac{u}{2}-\frac{u}{u - 4}&=\frac{u(u - 4)}{2(u - 4)}-\frac{2u}{2(u - 4)}\\ &=\frac{u(u - 4)-2u}{2(u - 4)}\\ &=\frac{u^{2}-4u - 2u}{2(u - 4)}\\ &=\frac{u^{2}-6u}{2(u - 4)}\\ &=\frac{u(u - 6)}{2(u - 4)} \end{align*}$$

\]

Step 2: Simplify the denominator \(\frac{4}{u - 4}-\frac{25}{2}\)

The common denominator here is \(2(u - 4)\).
\[

$$\begin{align*} \frac{4}{u - 4}-\frac{25}{2}&=\frac{8}{2(u - 4)}-\frac{25(u - 4)}{2(u - 4)}\\ &=\frac{8-25(u - 4)}{2(u - 4)}\\ &=\frac{8-25u + 100}{2(u - 4)}\\ &=\frac{108-25u}{2(u - 4)} \end{align*}$$

\]

Step 3: Divide the simplified numerator by the simplified denominator

Dividing by a fraction is the same as multiplying by its reciprocal. So we have:
\[

$$\begin{align*} \frac{\frac{u(u - 6)}{2(u - 4)}}{\frac{108-25u}{2(u - 4)}}&=\frac{u(u - 6)}{2(u - 4)}\times\frac{2(u - 4)}{108-25u}\\ \end{align*}$$

\]
The \(2(u - 4)\) terms cancel out:
\[
\frac{u(u - 6)}{108-25u}
\]

Final Answer for Problem 17:

\(\boxed{\frac{u(u - 6)}{108 - 25u}}\)

Now let's solve problem 18: \(\frac{\frac{1}{5}+\frac{1}{3}}{\frac{25}{3}+\frac{a^{2}}{a + 1}}\)

Step 1: Simplify the numerator \(\frac{1}{5}+\frac{1}{3}\)

The common denominator is \(15\).
\[

$$\begin{align*} \frac{1}{5}+\frac{1}{3}&=\frac{3}{15}+\frac{5}{15}\\ &=\frac{8}{15} \end{align*}$$

\]

Step 2: Simplify the denominator \(\frac{25}{3}+\frac{a^{2}}{a + 1}\)

The common denominator is \(3(a + 1)\).
\[

$$\begin{align*} \frac{25}{3}+\frac{a^{2}}{a + 1}&=\frac{25(a + 1)}{3(a + 1)}+\frac{3a^{2}}{3(a + 1)}\\ &=\frac{25(a + 1)+3a^{2}}{3(a + 1)}\\ &=\frac{25a+25 + 3a^{2}}{3(a + 1)}\\ &=\frac{3a^{2}+25a + 25}{3(a + 1)} \end{align*}$$

\]

Step 3: Divide the simplified numerator by the simplified denominator

\[

$$\begin{align*} \frac{\frac{8}{15}}{\frac{3a^{2}+25a + 25}{3(a + 1)}}&=\frac{8}{15}\times\frac{3(a + 1)}{3a^{2}+25a + 25}\\ &=\frac{8\times3(a + 1)}{15(3a^{2}+25a + 25)}\\ &=\frac{8(a + 1)}{5(3a^{2}+25a + 25)} \end{align*}$$

\]

Final Answer for Problem 18:

\(\boxed{\frac{8(a + 1)}{5(3a^{2}+25a + 25)}}\)

Let's solve problem 19: \(\frac{\frac{x}{9}+\frac{4}{5}}{\frac{2}{3}+\frac{x}{5}}\)

Step 1: Simplify the numerator \(\frac{x}{9}+\frac{4}{5}\)

The common denominator is \(45\).
\[

$$\begin{align*} \frac{x}{9}+\frac{4}{5}&=\frac{5x}{45}+\frac{36}{45}\\ &=\frac{5x + 36}{45} \end{align*}$$

\]

Step 2: Simplify the denominator \(\frac{2}{3}+\frac{x}{5}\)

The common denominator is \(15\).
\[

$$\begin{align*} \frac{2}{3}+\frac{x}{5}&=\frac{10}{15}+\frac{3x}{15}\\ &=\frac{10 + 3x}{15} \end{align*}$$

\]

Step 3: Divide the simplified numerator by the simplified denominator

\[

$$\begin{align*} \frac{\frac{5x + 36}{45}}{\frac{3x + 10}{15}}&=\frac{5x + 36}{45}\times\frac{15}{3x + 10}\\ &=\frac{(5x + 36)\times15}{45(3x + 10)}\\ &=\frac{5x + 36}{3(3x + 10)} \end{align*}$$

\]

Final Answer for Problem 19:

\(\boxed{\frac{5x + 36}{3(3x + 10)}}\)

Let's solve problem 20: \(\frac{\frac{9}{m}-\frac{3}{m + 3}}{\frac{m - 1}{3}+\frac{m}{m + 3}}\)

Step 1: Simplify the numerator \(\frac{9}{m}-\frac{3}{m + 3}\)

The common denominator is \(m(m + 3)\).
\[

$$\begin{align*} \frac{9}{m}-\frac{3}{m + 3}&=\frac{9(m + 3)}{m(m + 3)}-\frac{3m}{m(m + 3)}\\ &=\frac{9(m + 3)-3m}{m(m + 3)}\\ &=\frac{9m+27 - 3m}{m(m + 3)}\\ &=\frac{6m + 27}{m(m + 3)}\\ &=\frac{3(2m + 9)}{m(m + 3)} \end{align*}$$

\]

Step 2: Simplify the denominator \(\frac{m - 1}{3}+\frac{m}{m + 3}\)

The common denominator is \(3(m + 3)\).
\[

$$\begin{align*} \frac{m - 1}{3}+\frac{m}{m + 3}&=\frac{(m - 1)(m + 3)}{3(m + 3)}+\frac{3m}{3(m + 3)}\\ &=\frac{(m - 1)(m + 3)+3m}{3(m + 3)}\\ &=\frac{m^{2}+3m - m - 3+3m}{3(m + 3)}\\ &=\frac{m^{2}+5m - 3}{3(m + 3)} \end{align*}$$

\]

Step 3: Divide the simplified numerator by the simplified denominator

\[

$$\begin{align*} \frac{\frac{3(2m + 9)}{m(m + 3)}}{\frac{m^{2}+5m - 3}{3(m + 3)}}&=\frac{3(2m + 9)}{m(m + 3)}\times\frac{3(m + 3)}{m^{2}+5m - 3}\\ &=\frac{9(2m + 9)}{m(m^{2}+5m - 3)} \end{align*}$$

\]

Final Answer for Problem 20:

\(\boxed{\frac{9(2m + 9)}{m(m^{2}+5m - 3)}}\)