Answer: \(x=-1\) or \(x=2\). The graph has midline \(y=-2\), maximum \(3\), and minimum \(-7\).

(a) Start by isolating the modulus term:

\(5|2x-1|+8=23\)

\(5|2x-1|=15\)

\(|2x-1|=3.\)

For \(|A|=3\), either \(A=3\) or \(A=-3\). Hence

\(2x-1=3\quad\text{or}\quad2x-1=-3.\)

This gives

\(x=2\quad\text{or}\quad x=-1.\)

(b) The graph is

\(y=5\sin x-2.\)

The amplitude is \(5\), and the vertical translation is \(-2\), so the midline is

\(y=-2.\)

The maximum value is

\(-2+5=3,\)

and the minimum value is

\(-2-5=-7.\)

For \(0^\circ\leqslant x\leqslant360^\circ\), useful points for the sketch are

\((0^\circ,-2),\ (90^\circ,3),\ (180^\circ,-2),\ (270^\circ,-7),\ (360^\circ,-2).\)

So the graph is one full sine wave, centred on \(y=-2\), with maximum \(3\) and minimum \(-7\).