Sketch, on the same axes, the graphs of $y=5-\dfrac{1}{x-1}$ and $y=\dfrac{1}{(x-5)^2}$. Solve the inequality $5-\dfrac{1}{x-1}<\dfrac{1}{(x-5)^2}$.

1. Identify the asymptotes and intercepts of $y = 5 - \dfrac{1}{x-1}$.

Vertical asymptote: $x - 1 = 0 \implies x = 1$
Horizontal asymptote: As $x \to \pm\infty, y \to 5$, so $y = 5$

$y$-intercept: Let $x = 0 \implies y = 5 - \dfrac{1}{-1} = 6$. Point: $(0, 6)$
$x$-intercept: Let $y = 0 \implies 5 = \dfrac{1}{x-1} \implies x - 1 = 0.2 \implies x = 1.2$. Point: $(1.2, 0)$

2. Identify the asymptotes and intercepts of $y = \dfrac{1}{(x-5)^2}$.

Vertical asymptote: $x - 5 = 0 \implies x = 5$
Horizontal asymptote: As $x \to \pm\infty, y \to 0$, so $y = 0$

$y$-intercept: Let $x = 0 \implies y = \dfrac{1}{(-5)^2} = \dfrac{1}{25} = 0.04$. Point: $(0, 0.04)$
$x$-intercept: None, as $y > 0$ for all $x \neq 5$.

3. Sketch both graphs on the same axes.

Using a graphing calculator or the identified features, draw both curves on the same Cartesian plane.

• Include dotted lines for the asymptotes $x=1$, $x=5$, and $y=5$.
• Plot and label the intercepts at $(0,6)$, $(1.2,0)$, and $(0,0.04)$.
• Ensure the curve $y=5-\dfrac{1}{x-1}$ is drawn in the appropriate quadrants relative to its asymptotes (top-left and bottom-right).
• Ensure the curve $y=\dfrac{1}{(x-5)^2}$ approaches $x=5$ asymptotically in the positive $y$-direction from both sides.

1. Find the points of intersection of the two graphs using a graphing calculator.

Let $y_1 = 5-\dfrac{1}{x-1}$ and $y_2 = \dfrac{1}{(x-5)^2}$.
Using a graphing calculator to find the intersections of $y_1$ and $y_2$, we obtain the $x$-coordinates:
$x \approx 1.2028$
$x \approx 4.5422$
$x \approx 5.4578$

2. Determine the range of $x$ where $y_1 < y_2$ by referring to the sketched graphs.

The inequality $5-\dfrac{1}{x-1} < \dfrac{1}{(x-5)^2}$ holds where the graph of $y_1$ lies strictly below the graph of $y_2$.
From the sketch in part (a), and observing the vertical asymptotes at $x = 1$ and $x = 5$:

• For $1 < x < 1.2028$, the curve $y_1$ approaches $-\infty$ and lies below $y_2$.
• For $4.5422 < x < 5$, the curve $y_2$ approaches $+\infty$ and lies above $y_1$.
• For $5 < x < 5.4578$, $y_2$ descends from $+\infty$ and still lies above $y_1$.

3. State the final solution set.

Rounding the intersection values to 3 significant figures, the solution is:
$1 < x < 1.20 \text{ or } 4.54 < x < 5 \text{ or } 5 < x < 5.46$