Exercise $2.$ For $2x,$ it is true that

$\frac{1}{x}\frac{1}{x}\sqrt[\phantom{2}]{1\frac{1}{x^4}}$

Let's work through the proof to see why $(1\mathrm{.}1)$ is true. We will start with the assumption that $2x.$ We then use a more simple, true inequality, along with a step$-$by$-$step process, to build up to $(1\mathrm{.}1).$

$(1)$ To start, we know that when $2x$ we must have Why$?$ Convince yourself!$)$

$(2)$ adding $1$ to both sides of $0<mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfrac>$ we obtain $$

$(3)$ after taking the square root of both sides of your answer in $(2.)($since they're positive $($why$?)),$ we have $$

$(4)$ Now multiply both sides of your answer in $(3.)$ by $\frac{1}{x},$ which is positive $($why$?)$ to obtain the desired result $$

$1$