Describe how you know if an inequality is an “and” or an “or” problem. (use -3 < x+5 < 8). Also, how do you know if an absolute value question is an “and” or an “or” problem? Lastly, will you always have two solutions to an absolute value problem?
clearly the compound inequality is an and problem.
As for absolute value problems, think of the shape of the graph. It is a V shape. So, if |f(x)| < c, you have one interval, below the line y=c.
If |f(x)| > c, then you have the intervals outside the V, making it an or problem.
Of course, there may be other wrinkles, depending on the exact nature of f(x), but that's the idea.
To determine whether an inequality is an "and" or an "or" problem, you need to examine the connecting word (or words) between the inequalities.
For the inequality -3 < x+5 < 8:
1. If the connecting word is "AND," the inequality is an "and" problem. This means that both inequalities should be true for the overall statement to be true. In this case, there is no connecting word, so we can assume it is an "and" problem.
2. If the connecting word is "OR," the inequality is an "or" problem. This means that either one of the inequalities can be true for the overall statement to be true.
Now let's address absolute value questions:
To determine whether an absolute value question is an "and" or an "or" problem, you need to consider the structure of the question itself. If the question involves an "and" statement, it usually implies that both the expression inside the absolute value and its negative counterpart must meet certain criteria, making it an "and" problem. On the other hand, if the question involves an "or" statement, it typically implies that either the expression inside the absolute value or its negative counterpart must meet certain criteria, making it an "or" problem.
Regarding the number of solutions in an absolute value problem, there can be either one or two solutions. This depends on whether the expression within the absolute value evaluates to a positive or negative value.
For example, if the expression inside the absolute value is equal to a positive number, there will be two solutions because both the positive and negative versions of that number will satisfy the absolute value equation.
However, if the expression inside the absolute value equals zero or a negative number, there will be a single solution, as only the positive version is considered.
In summary, whether an inequality or an absolute value question is an "and" or "or" problem can be determined by examining the connecting words or the structure of the question, as well as the presence or absence of absolutes. Additionally, the number of solutions in an absolute value problem can vary based on the evaluation of the expression inside the absolute value.