Exponential equations and inequalities

The graph of the function $f(x)=a^x+b$ ( $a>0$, $a\ne 1$ ) has been moved $4$ units to the right and two units down. The shifted graph intersects the $x$-axis at the point $[4;0]$ and passes through the point $[8;3]$. Find $a$ and $b$ and solve the inequality $f(x)\le 5$.

The patient took a single dose of $50\mathrm{mg}$ of the drug. Within $3$ hours $40\mathrm{\%}$ of the dose was excreted from his body. The mass $m$ (mg) of the drug in the body after time $t$ (hours) is given by the formula $m(t)=m_0{a}^t$, where $m_0$ (mg) is the initial mass and $a$ is a constant. Calculate how much medicine the patient had in his body after $12$ hours.

Solve the following inequality. $$2^{x-2}-2^{x+1}-2^{x+4}\le -142$$

Solve the following inequality. $${\left(\frac{1}{2}\right)}^{2x}-7\cdot {\left(\frac{1}{2}\right)}^x-8<0$$