Problem 147 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can check inequalities by substituting 0 for the variable: When 0 belongs to the solution set, I should obtain a true statement, and when 0 does not belong to the solution set, I should obtain a false statement.

Step-by-Step Solution

Answer

The statement does not make absolute sense because substituting 0 into the variable of an inequality does not always yield a definitive conclusion regarding solution sets of the inequality.

Step 1: Understanding The Concept

An inequality expresses the idea that one quantity is less than or greater than another. A solution to the inequality is a number that, when substituted into the inequality, makes the inequality a true statement.

Step 2: Interpreting The Statement

In the statement, it mentions that inequalities can be checked by substituting 0 for the variable. It means, regardless of the given inequality, we always plug 0 into the variable to check if it's a viable solution or not.

Step 3: Testing the Statement

This statement is not entirely true. While it's a common technique to test 0 as it often simplifies the equation, it's not a definitive test whether 0 belongs in the solution set or not. An inequality might have a solution set that does not include 0, or it might not have a real solution at all. For example, for inequality \( x > 0 \), 0 is not a solution and for \( x^2 + 1 > 0 \), the inequality is true for all real numbers, hence always a true statement no matter the value of x.

Step 4: Conclusion

Therefore, while substituting 0 can be a useful technique at times, but it can't be used as a comprehensive method to determine the correctness of every inequalities. Consequently, it doesn’t make complete sense.