Question

$\|3x-9\|+2<2$

Answer 1

The solution set is the empty set, denoted by $\emptyset$ or {}.

Answer 2

The given inequality is $\|3x-9\|+2<2$.

To solve for $x$, we first isolate the absolute value term: Subtract 2 from both sides of the inequality: $\|3x-9\|+2 - 2 < 2 - 2$ $\|3x-9\| < 0$

The absolute value of any real number is always non-negative. That is, for any real number $y$, $\|y\| \ge 0$. In this case, $y = 3x-9$. So, $\|3x-9\|$ must be greater than or equal to 0. $\|3x-9\| \ge 0$

The inequality we need to satisfy is $\|3x-9\| < 0$. This requires the absolute value of $3x-9$ to be strictly less than zero. However, based on the property of absolute values, $\|3x-9\|$ can never be negative; it can only be zero or positive. Therefore, there is no real number $x$ for which $\|3x-9\| < 0$.

The set of values of $x$ that satisfy the inequality is the empty set.