Final answer:
The best first step in solving the equation is to simplify and find a common denominator. The solutions for x are 0.0216 and -0.0224.
Step-by-step explanation:
The best first step in solving the equation 5/x + (x+5)/(x+2) = -11/(x^2+2x) is to simplify the equation by finding a common denominator for all the fractions. In this case, the common denominator is x(x+2)(x^2+2x).
Multiplying each term in the equation by the common denominator will eliminate the fractions. After simplifying and rearranging the equation, it becomes a quadratic equation. Solving the quadratic equation will give you the solutions for x.
The two possible solutions for x are x = 0.0216 and x = -0.0224.
The student's question involves solving a rational equation with variable denominators. The best first step in solving the equation 5/x + (x+5)/(x+2) = -11/(x^2+2x) is to find a common denominator for all the terms so they can be combined into a single rational expression. Since we're dealing with a quadratic equation, we can expect to use the quadratic formula to find the solutions for x.
The process begins by expressing all denominators as factors of the least common denominator (LCD). Here, the LCD would be x(x+2). Once we multiply each term by the LCD, we can eliminate the fractions and simplify the equation. The equation then transforms into a quadratic equation, usually in the form ax^2 + bx + c = 0. From there, we use the quadratic formula, x = (-b ± √(b^2-4ac))/(2a), to solve for x, which may yield two solutions.
Taking into account the values mentioned in the question, x = 0.0216 or x = -0.0224, we would substitute these into the original equation to verify which, if any, makes sense in the context of the problem. One value might be rejected based on additional information, leaving us with the most sensible solution.
In other situations, when solving equations involved in chemical concentrations, as the later parts of the provided fragments suggest, one of the solutions might not be physically plausible. Furthermore, approximations can be made when the value of x is considerably smaller than other values in the problem, which simplifies the equation. However, this approximation seems less relevant to the given question and more to a problem in the field of chemistry or physics.