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?
first of all let's simplify the example
-3 < x+5 < 8
subtract 5 from each part
-8 < x < 3
In this notation, a < x < b, a is usually less than b
and x is any value "between" them.
As a matter of fact, I recall one text actually calling this the "between format", since we can read it as:
x lies between a and b
So the word "and" is always implied here
x > a AND x < b
if we were to say :
x > a OR x < b, then the entire number line would have to be included, which of course would not be true.
As to something like
|x| < 5
we have x < 5 ?? x > -5
Again, if we place OR between them, we get all the values of x
It must be AND to get the values of x between -5 and +5
In general, for
|stuff| < a, we have stuff < a AND stuff > -a, which is the same as:
-a < stuff < a
for
|stuff| > a , we have
stuff < -a OR stuff > a
If a > 0 , I cannot think of a case when there would not be two solutions.
If a < 0 , something like |stuff| < a would not make any sense, and there would be no solution.
To determine whether an inequality problem is an "and" or an "or" problem, you need to observe the connecting words or phrases used in the problem.
For example, consider the inequality -3 < x+5 < 8. In this case, we have the inequality -3 < x+5 and x+5 < 8. The word "and" is used to connect these two inequalities. This means that both inequalities need to be satisfied simultaneously, making it an "and" problem.
In the case of an absolute value question, determining whether it is an "and" or an "or" problem depends on the inequality sign outside the absolute value expression.
If the inequality sign is "<" or ">", it indicates that the absolute value expression must be less than or greater than the given value. For example, consider |x| < 5. In this case, the absolute value of x must be less than 5.
If the inequality sign is "<=" or ">=", it indicates that the absolute value expression must be less than or equal to, or greater than or equal to the given value. For example, consider |x| >= 3. In this case, the absolute value of x must be greater than or equal to 3.
In both cases, the absolute value question is an "or" problem. The reason is that the inequality sign outside the absolute value expression allows for the value on the absolute value expression to be equal to the given value as well.
Regarding your last question, when solving an absolute value problem, it is not always guaranteed to have two solutions. In some cases, there may be only one solution or even no solution. The number of solutions depends on the specific equation or inequality and the values involved.