How many real value(s) of x satisfy the equation $|b| = |b \, \sin(4x)|$, where $x \in [0, 2\pi]$ and $b$ is not zero? - HSC - SSCE Mathematics Extension 1 - Question 6 - 2024 - Paper 1

The equation $|\sin(4x)| = 1$ implies:

$\sin(4x) = 1 \quad \text{or} \quad \sin(4x) = -1$

This occurs at the angles:

• For $\sin(4x) = 1$: $4x = \frac{\pi}{2} + 2k\pi$, where $k$ is an integer.
• For $\sin(4x) = -1$: $4x = \frac{3\pi}{2} + 2k\pi$, where $k$ is an integer.

We now need to solve these for $x$ and find the values within the interval $[0, 2\pi]$.

1. From $4x = \frac{\pi}{2} + 2k\pi$:

   • $x = \frac{\pi}{8} + \frac{k\pi}{2}$
   • For $k = 0, 1$: $x = \frac{\pi}{8}, \frac{5\pi}{8}$ (within $[0, 2\pi]$)

2. From $4x = \frac{3\pi}{2} + 2k\pi$:

   • $x = \frac{3\pi}{8} + \frac{k\pi}{2}$
   • For $k = 0, 1$: $x = \frac{3\pi}{8}, \frac{7\pi}{8}$ (within $[0, 2\pi]$)

Combining these values gives us four distinct solutions: $\frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}$.

Thus, the total number of real values of $x$ that satisfy the equation is 4.