5. (a) On the same axes, sketch the graphs of \(y = x + 2\) and \(y = x ^ { 2 } - x - 6\) showing the coordinates of all points at which each graph crosses the coordinate axes.
(b) On your sketch, show, by shading, the region \(R\) defined by the inequalities $$y < x + 2 \text { and } y > x ^ { 2 } - x - 6$$ (c) Hence, or otherwise, find the set of values of \(x\) for which \(x ^ { 2 } - 2 x - 8 < 0\)
\includegraphics[max width=\textwidth, alt={}, center]{2217be5e-8edd-413f-9c97-212e585ff58d-10_921_1287_699_260} \includegraphics[max width=\textwidth, alt={}, center]{2217be5e-8edd-413f-9c97-212e585ff58d-11_2260_48_313_37} Quadratic: \(y = x ^ { 2 } - x - 6 = ( x - 3 ) ( x + 2 )\) $$x = 3 , \quad x = - 2 @ y = 0$$ Linear: \(\quad y = x + 2\) $$\begin{array} { l l } x = 0 : & y = 2 \quad ( 0,2 )
y = 0 : & x = - 2 \quad ( - 2,0 ) \end{array}$$ c) \(\quad x ^ { 2 } - 2 x - 8 < 0\) $$\begin{aligned} \therefore ( x - 4 ) ( x + 2 ) & < 0
x = 4 \quad x = - 2 &
\therefore - 2 < x < 4 & < 4 \end{aligned}$$