Let's examine the point [tex]\((1, -5)\)[/tex] in relation to the inequalities:
[tex]\[
\begin{array}{l}
y \leq 3x + 2 \\
y > -2x - 3
\end{array}
\][/tex]
### Step-by-Step Explanation:
1. First Inequality: [tex]\( y \leq 3x + 2 \)[/tex]
- Substitute the point [tex]\((1, -5)\)[/tex] into the inequality.
[tex]\[
y \leq 3x + 2
\][/tex]
- Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -5 \)[/tex]:
[tex]\[
-5 \leq 3(1) + 2
\][/tex]
- Simplify the right-hand side:
[tex]\[
-5 \leq 3 + 2
\][/tex]
[tex]\[
-5 \leq 5
\][/tex]
- This inequality is true, since [tex]\(-5\)[/tex] is indeed less than or equal to [tex]\(5\)[/tex].
Therefore, the point [tex]\((1, -5)\)[/tex] satisfies the inequality [tex]\( y \leq 3x + 2 \)[/tex].
2. Second Inequality: [tex]\( y > -2x - 3 \)[/tex]
- Substitute the point [tex]\((1, -5)\)[/tex] into the inequality.
[tex]\[
y > -2x - 3
\][/tex]
- Substitute [tex]\( x = 1 \)[/tex] and [tex]\( y = -5 \)[/tex]:
[tex]\[
-5 > -2(1) - 3
\][/tex]
- Simplify the right-hand side:
[tex]\[
-5 > -2 - 3
\][/tex]
[tex]\[
-5 > -5
\][/tex]
- This inequality is false, since [tex]\(-5\)[/tex] is not greater than [tex]\(-5\)[/tex] (they are equal).
Therefore, the point [tex]\((1, -5)\)[/tex] does not satisfy the inequality [tex]\( y > -2x - 3 \)[/tex].
### Graphical Explanation:
To understand this graphically, let's plot both inequalities on the coordinate plane.
1. Plotting [tex]\( y \leq 3x + 2 \)[/tex]:
- The boundary line is [tex]\( y = 3x + 2 \)[/tex], which is a line with slope [tex]\(3\)[/tex] and y-intercept [tex]\(2\)[/tex].
- Since the inequality is [tex]\( \leq \)[/tex], we shade the region below (including) the line.
2. Plotting [tex]\( y > -2x - 3 \)[/tex]:
- The boundary line is [tex]\( y = -2x - 3 \)[/tex], which is a line with slope [tex]\(-2\)[/tex] and y-intercept [tex]\(-3\)[/tex].
- Since the inequality is [tex]\( > \)[/tex], we shade the region above (excluding) the line.
Now, by plotting the point [tex]\((1, -5)\)[/tex]:
- For [tex]\( y \leq 3x + 2 \)[/tex], check if the point is in the shaded region below the line [tex]\( y = 3x + 2 \)[/tex]. It is, so it satisfies this inequality.
- For [tex]\( y > -2x - 3 \)[/tex], check if the point is in the shaded region above the line [tex]\( y = -2x - 3 \)[/tex]. It isn't, as it lies on the boundary line itself.
### Conclusion:
The point [tex]\((1, -5)\)[/tex] satisfies the inequality [tex]\( y \leq 3x + 2 \)[/tex] but does not satisfy the inequality [tex]\( y > -2x - 3 \)[/tex]. Thus, in relation to the given system of inequalities, the point satisfies only the first inequality but not the second.
