Final answer:
The solution to the inequality involves turning it into an equation set to zero and using the quadratic formula to find possible values of x. Then, using the interval test, we determine which solutions satisfy the original inequality. Additional simplifications may occur if one variable is much smaller than others.
Step-by-step explanation:
The student has presented a complex mathematical problem requiring the solution of an inequality. This involves various steps including distributing terms, combining like terms, applying arithmetic properties, and solving quadratic equations. Although the question references various equations and methods such as completing the square, the quadratic equation, and making assumptions for small values of x, the exact inequality to be solved is not clearly stated. Nevertheless, in general, solving such inequalities typically involves turning the inequality into an equation that equals zero, and then factoring or using the quadratic formula to find the possible values of x.
Once the potential solutions are found using the quadratic formula (x = [-b ± √(b² - 4ac)]/(2a)), we must then determine which of these solutions satisfy the original inequality. This is often done by testing intervals between the solutions, a method known as the interval test.
In some cases, as exemplified by one of the provided snippets, if x is very small compared to another number in the equation (e.g., x << 0.200), it may be approximated as negligible, simplifying the process of solving. However, such approximations should be used with caution as they might not always yield an accurate answer.