Problem source

Show that 
\[
x^2-y^2 +x+3y-2 = (x-y+2)(x+y-1)
\]
and hence, or otherwise, indicate by means of a
 sketch the region of the $x$-$y$ plane for which
$$
x^2-y^2 +x+3y>2.
$$
Sketch also the region of the $x$-$y$ plane for which
$$
x^2-4y^2 +3x-2y<-2.
$$
Give the coordinates of 
a point for which both inequalities are satisfied or explain why
no such point exists.

Solution source

\begin{align*}
&& (x-y+2)(x+y-1) &= (x-y)(x+y)-(x-y)+2(x+y)-2 \\
&&&= x^2-y^2+x+3y-2
\end{align*}

\begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){2*(#1)*((#1)^2 - 5)/((#1)^2-4)};
    \def\xl{-5};
    \def\xu{5};
    \def\yl{-5};
    \def\yu{5};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);

        \draw[thick, blue] (\xl, {\xl+2}) -- (\xu, {\xu+2});
        \draw[thick, blue] (\xl, {1-\xl}) -- (\xu, {1-\xu});
        % \draw[thick, blue] (\xl, {0.25*\xl}) -- (\xu, {0.25*\xu});

        \filldraw[blue, opacity = 0.2] (\xl, {\xl+2}) -- (\xl, {1-\xl}) -- ({-1/2},{3/2}) -- cycle;
        \filldraw[blue, opacity = 0.2] (\xu, {\xu+2}) -- (\xu, {1-\xu}) -- ({-1/2},{3/2}) -- cycle;
        % \filldraw[blue, opacity = 0.2] (\xu, {0.25*\xu}) -- (\xu, {-\xu}) -- (0,0) -- cycle;
        
    \end{scope}
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
    
    \end{tikzpicture}
\end{center}

\begin{align*}
x^2-4y^2 +3x-2y+2 &= (x - 2 y + 1) (x + 2 y + 2)
\end{align*}

\begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){2*(#1)*((#1)^2 - 5)/((#1)^2-4)};
    \def\xl{-5};
    \def\xu{5};
    \def\yl{-5};
    \def\yu{5};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);

        % \draw[thick, blue] (\xl, {\xl+2}) -- (\xu, {\xu+2});
        % \draw[thick, blue] (\xl, {1-\xl}) -- (\xu, {1-\xu});

        
        \draw[thick, red] (\xl, {(\xl+1)/2}) -- (\xu, {(\xu+1)/2});
        \draw[thick, red] (\xl, {-(\xl+2)/2}) -- (\xu, {-(\xu+2)/2});
        % \draw[thick, blue] (\xl, {0.25*\xl}) -- (\xu, {0.25*\xu});

        % \filldraw[blue, opacity = 0.2] (\xl, {\xl+2}) -- (\xl, {1-\xl}) -- ({-1/2},{3/2}) -- cycle;
        % \filldraw[blue, opacity = 0.2] (\xu, {\xu+2}) -- (\xu, {1-\xu}) -- ({-1/2},{3/2}) -- cycle;
        \filldraw[red, opacity = 0.2] (\xl, {(\xl+1)/2}) -- (\xu, {-(\xu+2)/2}) -- ({-3/2},{-1/4}) -- cycle;
        \filldraw[red, opacity = 0.2] (\xu, {(\xu+1)/2}) -- (\xl, {-(\xl+2)/2}) -- ({-3/2},{-1/4}) -- cycle;
        
    \end{scope}
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
    
    \end{tikzpicture}
\end{center}


\begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){2*(#1)*((#1)^2 - 5)/((#1)^2-4)};
    \def\xl{-5};
    \def\xu{5};
    \def\yl{-5};
    \def\yu{5};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);

        \draw[thick, blue] (\xl, {\xl+2}) -- (\xu, {\xu+2});
        \draw[thick, blue] (\xl, {1-\xl}) -- (\xu, {1-\xu});

        
        \draw[thick, red] (\xl, {(\xl+1)/2}) -- (\xu, {(\xu+1)/2});
        \draw[thick, red] (\xl, {-(\xl+2)/2}) -- (\xu, {-(\xu+2)/2});
        % \draw[thick, blue] (\xl, {0.25*\xl}) -- (\xu, {0.25*\xu});

        \filldraw[blue, opacity = 0.2] (\xl, {\xl+2}) -- (\xl, {1-\xl}) -- ({-1/2},{3/2}) -- cycle;
        \filldraw[blue, opacity = 0.2] (\xu, {\xu+2}) -- (\xu, {1-\xu}) -- ({-1/2},{3/2}) -- cycle;
        \filldraw[red, opacity = 0.2] (\xl, {(\xl+1)/2}) -- (\xu, {-(\xu+2)/2}) -- ({-3/2},{-1/4}) -- cycle;
        \filldraw[red, opacity = 0.2] (\xu, {(\xu+1)/2}) -- (\xl, {-(\xl+2)/2}) -- ({-3/2},{-1/4}) -- cycle;
        
    \end{scope}
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
    
    \end{tikzpicture}
\end{center}

Consider the point $x = 0, y = \frac32$, then $\frac92 - \frac94 = \frac94 > 2$ and $-4\cdot\frac94-2\cdot \frac32 = -12 < -2$ so this is an example of a point in both regions