Find the set of all z such that neither 2+z nor 2-z is in the interval (-1,1]. Express your answer as an interval or as a union of intervals.
Well, well, well, looks like we have a mathematical riddle on our hands! Let's see if we can solve this.
We're looking for values of 'z' that make both 2+z and 2-z fall outside the interval (-1,1]. In other words, we need to find values of 'z' that make both of these expressions less than or equal to -1, or greater than 1.
So, let's break it down step by step:
First, let's solve 2+z ≤ -1. By subtracting 2 from both sides, we get z ≤ -3.
Secondly, let's solve 2-z ≤ -1. By subtracting 2 from both sides, we get -z ≤ -3, and by multiplying both sides by -1 (to flip the inequality sign), we find z ≥ 3.
Now, we have two inequalities: z ≤ -3 and z ≥ 3. This means that the set of 'z' that satisfy both conditions is the empty set. That's right, we've got no solution!
In interval notation, we express the empty set as ∅. So the answer is ∅.
Hope that brings a smile to your face, even if it's not the solution you were hoping for!
To find the set of all z such that neither 2+z nor 2-z is in the interval (-1,1], we need to consider both cases separately.
Case 1: 2+z is not in (-1,1]
To satisfy this condition, we need to find the values of z for which 2+z is outside the interval (-1,1].
If 2+z is outside the interval (-1,1], it means that either 2+z < -1 or 2+z > 1.
For 2+z < -1:
Solving the inequality 2+z < -1 for z, we get:
z < -3
For 2+z > 1:
Solving the inequality 2+z > 1 for z, we get:
z > -1
Therefore, the values of z that satisfy neither 2+z < -1 nor 2+z > 1 are z < -3 or z > -1.
Case 2: 2-z is not in (-1,1]
To satisfy this condition, we need to find the values of z for which 2-z is outside the interval (-1,1].
If 2-z is outside the interval (-1,1], it means that either 2-z < -1 or 2-z > 1.
For 2-z < -1:
Solving the inequality 2-z < -1 for z, we get:
z > 3
For 2-z > 1:
Solving the inequality 2-z > 1 for z, we get:
z < 1
Therefore, the values of z that satisfy neither 2-z < -1 nor 2-z > 1 are z > 3 or z < 1.
Combining the results from both cases, we have:
z < -3 or z > -1 and z > 3 or z < 1
We can express this as a union of intervals as:
(-∞, -3) ∪ (-1, 1) ∪ (3, +∞)
So, the set of all z such that neither 2+z nor 2-z is in the interval (-1,1] can be represented as the union of the intervals (-∞, -3), (-1, 1), and (3, +∞).
To find the set of all z such that neither 2+z nor 2-z is in the interval (-1,1], we can break it down into two separate conditions.
First, let's consider the inequality 2+z not being in the interval (-1, 1]. To do this, we can set up the inequality and solve for z:
2 + z <= -1 or 2 + z > 1
Solving these inequalities separately, we get:
z <= -3 or z > -1
This means that z can be any value less than or equal to -3, or any value greater than -1.
Secondly, let's consider the inequality 2-z not being in the interval (-1, 1]. We can set up the inequality and solve for z:
2 - z <= -1 or 2 - z > 1
Solving these inequalities separately, we get:
z >= 3 or z < 1
This means that z can be any value greater than or equal to 3, or any value less than 1.
Combining the two sets of solutions, we find that z can be any value less than or equal to -3, any value greater than -1, any value greater than or equal to 3, or any value less than 1.
Expressing this as an interval or a union of intervals, we can write the final answer as:
(-∞, -3] ∪ (-1, ∞) ∪ [3, ∞) ∪ (-∞, 1)
If z>=0,
2+z > 1 -- z > -1
2-z <= -1 -- z >= 3
If z<0,
2-z > 1 -- z < 1
2+z <= -1 -- z <= -3
graph those on the number line, and you will find your answer.
Be sure to check a few values to make sure they work, especially at the boundaries.