求满足 $|3^{2x-1}-4|>2$ 的 $x$ 的取值集合。答案保留 $3$ 位有效数字。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

绝对值大于 $2$ 表示内部量小于 $-2$ 或大于 $2$。指数函数递增，取对数时不改变不等号方向。

The inequality gives $3^{2x-1}-4<-2$ or $3^{2x-1}-4>2$. Thus

Taking logarithms gives

Since $\ln3>0$, the exact ranges are

The endpoints are $0.815464\ldots$ and $1.315464\ldots$. To three significant figures, the answer is $x<0.815$ or $x>1.32$.

也可以令指数项为一个正变量，将绝对值不等式平方后分解。求出该变量的范围，再还原为 $x$。

Let $w=3^{2x-1}>0$. Squaring the original nonnegative sides gives

Thus $0<w<2$ or $w>6$. Since $w$ increases with $x$, this gives the exact ranges

To three significant figures these are $x<0.815$ or $x>1.32$.