Final answer:
Jane's idea of substituting positive and negative values for a variable to validate an equation is incorrect since this does not prove the equation for all possible values. The approach also overlooks potential real-world constraints on the variable. Validity requires proof that holds under all instances, not just selected examples.
Step-by-step explanation:
Jane's idea involves attempting to prove the validity of an equation by testing it with both positive and negative values for a variable, n. This approach assumes that if the equation holds true for a sample value of n, then the equation is generally valid. However, this method is flawed because proving an equation requires a proof that is valid for all possible values, not just particular instances. For example, in situations where there are two possible values of x in a quadratic equation, one derived from using a + sign and the other from a - sign in the numerator, only one value may make sense in the given context. This is often the case in problems involving concentrations or physics where the negative value might be physically impossible, thus the positive value is the only meaningful solution.
Furthermore, in deductive reasoning, replacing the variables with specific statements to prove validity is akin to a disjunctive syllogism. While specific true statements may lead to a true conclusion, the validity of the argument form must hold in all cases, not just the chosen ones. Hence, a counterexample where the premises are true but the conclusion is false can prove an argument invalid, as in Example 9.
Finally, when solving problems, it's essential to always check if the answer is reasonable. For instance, in physics problems involving work and energy, signs of quantities such as work against friction or potential energy should be considered, and the physical plausibility of the outcomes should be evaluated to ensure they make sense in the real-world scenario, as highlighted in Step 6.