Question

If the domain of $h\left(x\right)=\ln \left(|2x-9|\right)$ is $(-\infty ,x)U(x,\infty )$.

Find x.

Answer 1

Since the graph of $y=|x|$ is a V-shaped graph which is symmetric to the y-axis and is always positive. The two lines forming the V originate at the origin $(0,0)$ and go at $45$ degree angles (one with a slope of $1$ and one with a slope of $-1$). Thus $x$ can have any real value so its domain is all the real numbers.

Now we put $2x-9=0$ that is $2x=9$ that is $x=\frac{9}{2}$

Therefore, $\ln |2x-9|$ is either positive or zero for $x=\frac{9}{2}$.

Hence, the domain of the given function is $(-infinity,\frac{9}{2})\cup (\frac{9}{2},+infinity)$.