Solve the inequality (2/5)^x < sqrt((4/25)^((3x + 2)/x)).

Step-by-step solution

1. 1

   State the domain first. The exponent $\frac{3x+2}{x}$ is undefined at $x=0$, so $x\ne 0$ is a standing restriction. Everything else is defined for all real $x$, since a positive base raised to any real power is positive.

2. 2

   Write both sides with the same base. Notice $\frac{4}{25}=\left(\frac{2}{5}\right)^2$, so the square root cancels the square:

   $\sqrt{\left(\frac{4}{25}\right)^{\frac{3x+2}{x}}}=\left[\left(\tfrac{2}{5}\right)^{2\cdot\frac{3x+2}{x}}\right]^{1/2}=\left(\frac{2}{5}\right)^{\frac{3x+2}{x}}$

   The inequality becomes a pure comparison of exponents on the same base:

   $\left(\frac{2}{5}\right)^{x}<\left(\frac{2}{5}\right)^{\frac{3x+2}{x}}$

3. 3

   Flip the inequality because the base is less than 1. The function $t\mapsto\left(\frac25\right)^t$ is decreasing, so a smaller value corresponds to a larger exponent:

   $x>\frac{3x+2}{x}$

   Forgetting this reversal is the single most common error in exponential inequalities with a fractional base.

4. 4

   Move everything to one side — never multiply by $x$. Multiplying across by $x$ would require splitting into cases for the sign of $x$; subtracting is safer:

   $x-\frac{3x+2}{x}>0\;\Longleftrightarrow\;\frac{x^2-3x-2}{x}>0$

5. 5

   Find the critical points and build a sign chart. The numerator vanishes when $x^2-3x-2=0$, i.e.

   $x=\frac{3\pm\sqrt{9+8}}{2}=\frac{3\pm\sqrt{17}}{2}\approx-0.5616\ \text{and}\ 3.5616$

   Together with the zero of the denominator, the critical points in order are $\frac{3-\sqrt{17}}{2}$, $0$, $\frac{3+\sqrt{17}}{2}$. Testing one point in each of the four intervals gives the signs $-,\;+,\;-,\;+$.

6. 6

   Write the solution set and spot-check it. The quotient is positive on

   $x\in\left(\frac{3-\sqrt{17}}{2},\,0\right)\cup\left(\frac{3+\sqrt{17}}{2},\,+\infty\right)$

   Check $x=-0.4$: left side $0.4^{-0.4}\approx1.44$, right side $0.4^{(3(-0.4)+2)/(-0.4)}=0.4^{-2}=6.25$ — the inequality holds. Check $x=1$: $0.4<0.4^{5}=0.01$ is false, as the chart predicts.