Sure, let's go through the conditions and check each of the given points step by step.
We need to verify each point ([tex]\( x, y \)[/tex]) against the following conditions:
1. [tex]\( y \leq -x + 1 \)[/tex]
2. [tex]\( y > x \)[/tex]
Let's evaluate each point:
1. Point [tex]\((-3, 5)\)[/tex]
- Check if [tex]\( 5 \leq -(-3) + 1 \)[/tex]
- Calculation: [tex]\( 5 \leq 3 + 1 \)[/tex]
- Simplifies to: [tex]\( 5 \leq 4 \)[/tex] (False)
- Check if [tex]\( 5 > -3 \)[/tex]
- Calculation: [tex]\( 5 > -3 \)[/tex] (True)
While [tex]\( y > x \)[/tex] is true, [tex]\( y \leq -x + 1 \)[/tex] is false. So, this point does not satisfy both conditions.
2. Point [tex]\((-2, 2)\)[/tex]
- Check if [tex]\( 2 \leq -(-2) + 1 \)[/tex]
- Calculation: [tex]\( 2 \leq 2 + 1 \)[/tex]
- Simplifies to: [tex]\( 2 \leq 3 \)[/tex] (True)
- Check if [tex]\( 2 > -2 \)[/tex]
- Calculation: [tex]\( 2 > -2 \)[/tex] (True)
Both conditions are true for this point. So, it satisfies both conditions.
3. Point [tex]\((-1, -3)\)[/tex]
- Check if [tex]\( -3 \leq -(-1) + 1 \)[/tex]
- Calculation: [tex]\( -3 \leq 1 + 1 \)[/tex]
- Simplifies to: [tex]\( -3 \leq 2 \)[/tex] (True)
- Check if [tex]\( -3 > -1 \)[/tex]
- Calculation: [tex]\( -3 > -1 \)[/tex] (False)
While [tex]\( y \leq -x + 1 \)[/tex] is true, [tex]\( y > x \)[/tex] is false. So, this point does not satisfy both conditions.
4. Point [tex]\((0, -1)\)[/tex]
- Check if [tex]\( -1 \leq -(0) + 1 \)[/tex]
- Calculation: [tex]\( -1 \leq 0 + 1 \)[/tex]
- Simplifies to: [tex]\( -1 \leq 1 \)[/tex] (True)
- Check if [tex]\( -1 > 0 \)[/tex]
- Calculation: [tex]\( -1 > 0 \)[/tex] (False)
While [tex]\( y \leq -x + 1 \)[/tex] is true, [tex]\( y > x \)[/tex] is false. So, this point does not satisfy both conditions.
After examining all the points, we find that the only point that meets both conditions is [tex]\((-2, 2)\)[/tex].
So the valid point is:
[tex]\[ [(-2, 2)] \][/tex]
