Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Get immediate and reliable answers to your questions from a community of experienced professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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.
[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.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
