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\begin{center}
    \Large\textbf{Further Pure 1}\\
    \large Chapter 4 Inequalities with Graph Sketching Practice
\end{center}
\vspace{0.cm}

\section*{Easier: Linear and Simple Rational Functions}

\begin{enumerate}[label=\textbf{\arabic*.}]
    \item 
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item On the same set of axes, sketch the graphs of $y = x$ and $y = \frac{4}{x}$.
        \item Find the coordinates of the points of intersection of $y = x$ and $y = \frac{4}{x}$.
        \item Hence write down the solution to the inequality $x \ge \frac{4}{x}$.
    \end{enumerate}

    \vspace{0.2cm}
    \item $f(x) = x + 2$ and $g(x) = \frac{3}{x}$, $x \neq 0$.
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item Sketch $y = f(x)$ and $y = g(x)$ on the same set of axes.
        \item Solve $f(x) = g(x)$.
        \item Hence, or otherwise, solve the inequality $f(x) < g(x)$.
    \end{enumerate}

    \vspace{0.2cm}
    \item 
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item On the same set of axes, sketch the graphs of $y = \frac{2}{x}$ and $y = \frac{1}{x-2}$.
        \item Find the points of intersection of the two graphs.
        \item Write down the solution to the inequality $\frac{2}{x} > \frac{1}{x-2}$.
    \end{enumerate}

    \vspace{0.2cm}
    \item 
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item On the same set of axes, sketch the graphs of $y = 2x - 1$ and $y = \frac{2}{x-2}$.
        \item Find the exact coordinates of the points of intersection of $y = 2x - 1$ and $y = \frac{2}{x-2}$.
        \item Solve the inequality $2x - 1 \le \frac{2}{x-2}$.
    \end{enumerate}

    \vspace{0.2cm}
    \item $f(x) = \frac{4}{x^2}$, $x \neq 0$ and $g(x) = x + 3$.
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item Sketch $y = f(x)$ and $y = g(x)$ on the same set of axes.
        \item Find the coordinates of the points where the graphs intersect.
        \item Hence write down the solution to the inequality $f(x) < g(x)$. Give your answer using set notation.
    \end{enumerate}
\end{enumerate}

\vspace{0.2cm}
\section*{Moderate: Rational vs. Rational (Linear Denominators)}

\begin{enumerate}[label=\textbf{\arabic*.}, resume]
    \item 
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item On the same set of axes, sketch the graphs of $y = \frac{x+1}{x-2}$ and $y = 3$.
        \item Find the point of intersection of $y = \frac{x+1}{x-2}$ and $y = 3$.
        \item Solve the inequality $\frac{x+1}{x-2} \ge 3$.
    \end{enumerate}

    \vspace{0.2cm}
    \item 
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item On the same set of axes, sketch the graphs of $y = \frac{2x}{x+1}$ and $y = x$.
        \item Find the points of intersection of $y = \frac{2x}{x+1}$ and $y = x$.
        \item Hence, or otherwise, solve the inequality $\frac{2x}{x+1} > x$.
    \end{enumerate}

    \vspace{0.3cm}
    \item $f(x) = \frac{2}{x-1}$, $x \neq 1$ and $g(x) = \frac{x}{x+2}$, $x \neq -2$.
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item Sketch $y = f(x)$ and $y = g(x)$ on the same set of axes.
        \item Solve $f(x) = g(x)$, hence write down the solution to the inequality $f(x) \le g(x)$.
    \end{enumerate}

    \vspace{0.2cm}
    \item 
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item On the same set of axes, sketch the graphs of $y = \frac{x}{x+3}$ and $y = \frac{2}{x+1}$.
        \item Find the exact coordinates of the points of intersection.
        \item Solve the inequality $\frac{x}{x+3} < \frac{2}{x+1}$. Give your answer using set notation.
    \end{enumerate}

    \vspace{0.2cm}
    \item 
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item On the same set of axes, sketch the graphs of $y = \frac{2(x-2)}{x+1}$ and $y = \frac{x-2}{x-1}$.
        \item Find the points of intersection of $y = \frac{2(x-2)}{x+1}$ and $y = \frac{x-2}{x-1}$.
        \item Write down the solution to the inequality $\frac{2(x-2)}{x+1} \ge \frac{x-2}{x-1}$.
    \end{enumerate}
\end{enumerate}

\vspace{0.2cm}
\section*{Harder: Squared Denominators, Complex Rational Expressions, and Challenge Problems}

\begin{enumerate}[label=\textbf{\arabic*.}, resume]
    \item $f(x) = \frac{3}{x^2}$, $x \neq 0$ and $g(x) = \frac{4-x}{x}$, $x \neq 0$.
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item Sketch $y = f(x)$ and $y = g(x)$ on the same set of axes.
        \item Solve $f(x) = g(x)$.
        \item Hence, or otherwise, solve the inequality $f(x) > g(x)$.
    \end{enumerate}

    \vspace{0.3cm}
    \item 
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item On the same set of axes, sketch the graphs of $y = \frac{4}{x^2-9}$ and $y = \frac{1}{x-3}$.
        \item Find the point of intersection of the two graphs.
        \item Solve the inequality $\frac{4}{x^2-9} \le \frac{1}{x-3}$.
    \end{enumerate}

    \vspace{0.3cm}
    \item 
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item On the same set of axes, sketch the graphs of $y = \frac{x}{4-x}$ and $y = \frac{4x}{(x-1)^2}$.
        \item Find the exact points of intersection of $y = \frac{x}{4-x}$ and $y = \frac{4x}{(x-1)^2}$.
        \item Hence, or otherwise, solve the inequality $\frac{x}{4-x} \ge \frac{4x}{(x-1)^2}$.
    \end{enumerate}

    \vspace{0.3cm}
    \item 
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item On the same set of axes, sketch the graphs of $y = x - 1$ and $y = \frac{4(1-x)}{(x+1)(x-4)}$.
        \item Find the points of intersection of $y = x - 1$ and $y = \frac{4(1-x)}{(x+1)(x-4)}$.
        \item Write down the solution to the inequality $x - 1 \le \frac{4(1-x)}{(x+1)(x-4)}$.
    \end{enumerate}

    \vspace{0.3cm}
    \item 
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item On the same set of axes, sketch the graphs of $y = \frac{x^2}{x-2}$ and $y = x + 3$. \textit{(Note: This is beyond the specification. This graph has an oblique asymptote-- these occur when the order of the numerator is greater than the order of the denominator. Divide $x^2$ by $x-2$ to find the oblique asymptote.)}.
        \item Find the point of intersection of the two graphs.
        \item Solve the inequality $\frac{x^2}{x-2} < x + 3$. Give your answer using set notation.
    \end{enumerate}
\end{enumerate}

\end{document}