Question

Number of integral values of 'x' satisfying the equation $3^{|x+1|}-2.3^x = 2.|3^x-1|+1$ is

Answer 1

2

Answer 2

The equation involves absolute values $|x+1|$ and $|3^x-1|$. We analyze the equation by considering intervals based on the critical points where the expressions inside the absolute values change sign. The critical points are $x=-1$ (from $x+1=0$) and $x=0$ (from $3^x-1=0$). We examine the equation in the intervals $x < -1$, $-1 \le x < 0$, and $x \ge 0$.

For $x < -1$, we get $|x+1| = -x-1$ and $|3^x-1| = 1-3^x$. Solving the resulting equation yields $x=-2$, which is in the interval.

For $-1 \le x < 0$, we get $|x+1| = x+1$ and $|3^x-1| = 1-3^x$. Solving the resulting equation yields $x=0$, which is not in this interval.

For $x \ge 0$, we get $|x+1| = x+1$ and $|3^x-1| = 3^x-1$. Solving the resulting equation yields $x=0$, which is in this interval.

The integral solutions are $x=-2$ and $x=0$. There are 2 integral solutions.

The final answer is $\boxed{2}$.