Let's solve the inequality [tex]\( y > -x - 5 \)[/tex] step by step and find some sample values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy this inequality.
First, we need to understand that this inequality describes a region on the coordinate plane. The boundary line of this region is given by the equation [tex]\( y = -x - 5 \)[/tex]. For any point above this line, the inequality [tex]\( y > -x - 5 \)[/tex] will be true.
Let's choose a few values for [tex]\( x \)[/tex] and then find the corresponding [tex]\( y \)[/tex] values that lie on the boundary line [tex]\( y = -x - 5 \)[/tex], but remember our values for [tex]\( y \)[/tex] need to be strictly greater than [tex]\( -x - 5 \)[/tex] to satisfy the inequality.
We'll start by finding the values on the boundary line:
1. When [tex]\( x = -2 \)[/tex]:
[tex]\[
y = -(-2) - 5 = 2 - 5 = -3
\][/tex]
2. When [tex]\( x = -1 \)[/tex]:
[tex]\[
y = -(-1) - 5 = 1 - 5 = -4
\][/tex]
3. When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 0 - 5 = -5
\][/tex]
4. When [tex]\( x = 1 \)[/tex]:
[tex]\[
y = -1 - 5 = -6
\][/tex]
5. When [tex]\( x = 2 \)[/tex]:
[tex]\[
y = -2 - 5 = -7
\][/tex]
Now, we will compile these values into a list:
- For [tex]\( x \)[/tex] values: [-2, -1, 0, 1, 2]
- Corresponding [tex]\( y \)[/tex] values on the boundary: [-3, -4, -5, -6, -7]
So, we have the points (-2, -3), (-1, -4), (0, -5), (1, -6), and (2, -7) lying exactly on the boundary line [tex]\( y = -x - 5 \)[/tex].
However, since we need [tex]\( y \)[/tex] values to be strictly greater than these boundary values to satisfy the inequality [tex]\( y > -x - 5 \)[/tex], the [tex]\( y \)[/tex] values should be a little higher than those listed on the boundary line. Therefore, any [tex]\( y \)[/tex] values greater than -3, -4, -5, -6, and -7 for the corresponding [tex]\( x \)[/tex] values of -2, -1, 0, 1, and 2 respectively would satisfy the inequality.
To convey this clearly:
- For [tex]\( x = -2 \)[/tex], [tex]\( y \)[/tex] should be greater than -3.
- For [tex]\( x = -1 \)[/tex], [tex]\( y \)[/tex] should be greater than -4.
- For [tex]\( x = 0 \)[/tex], [tex]\( y \)[/tex] should be greater than -5.
- For [tex]\( x = 1 \)[/tex], [tex]\( y \)[/tex] should be greater than -6.
- For [tex]\( x = 2 \)[/tex], [tex]\( y \)[/tex] should be greater than -7.
Thus, the points (-2, -3), (-1, -4), (0, -5), (1, -6), and (2, -7) mark the boundary, but any point above these will satisfy our inequality [tex]\( y > -x - 5 \)[/tex].
