Question
what are the plot points of this equation 10.5x + 5.5y + 150 is less than or equal to 300
no decimals or negative numbers allowed
no decimals or negative numbers allowed
Answers
GPT-4o mini
To analyze the inequality \(10.5x + 5.5y + 150 \leq 300\) under the constraint of no decimals or negative numbers, we can first simplify the inequality and then express it as a linear equation.
1. **Simplify the Inequality:**
\[
10.5x + 5.5y + 150 \leq 300
\]
Subtract 150 from both sides:
\[
10.5x + 5.5y \leq 150
\]
2. **Clear Decimals:**
To eliminate the decimals, multiply the entire inequality by 2:
\[
21x + 11y \leq 300
\]
3. **Find the Boundary Line:**
The equality corresponding to our inequality is:
\[
21x + 11y = 300
\]
We will find the x- and y-intercepts to help visualize the constraint.
- **To find the x-intercept** (set \(y = 0\)):
\[
21x = 300 \implies x = \frac{300}{21} \approx 14.29 \quad (\text{closest integer } x = 14)
\]
- **To find the y-intercept** (set \(x = 0\)):
\[
11y = 300 \implies y = \frac{300}{11} \approx 27.27 \quad (\text{closest integer } y = 27)
\]
Therefore, the intercepts are approximately:
- \(x\)-intercept at \( (14, 0) \)
- \(y\)-intercept at \( (0, 27) \)
4. **Determine Integer Coordinates:**
We can substitute integer values in the original equation to find valid \((x, y)\) pairs that satisfy the inequality. Given \(x\) and \(y\) must be non-negative integers and keep the total under 300.
Let's try some integer combinations while keeping \(21x + 11y \leq 300\):
- \(x = 0\):
- \(11y \leq 300 \implies y \leq 27\) → Pairs: \((0, 0), (0, 1), ..., (0, 27)\) (28 pairs)
- \(x = 1\):
- \(21(1) + 11y \leq 300 \implies 11y \leq 279 \implies y \leq 25\) → Pairs: \((1, 0), (1, 1), ..., (1, 25)\) (26 pairs)
- \(x = 2\):
- \(21(2) + 11y \leq 300 \implies 11y \leq 258 \implies y \leq 23\) → Pairs: \((2, 0), (2, 1), ..., (2, 23)\) (24 pairs)
- \(x = 3\):
- \(21(3) + 11y \leq 300 \implies 11y \leq 237 \implies y \leq 21\) → Pairs: \((3, 0), (3, 1), ..., (3, 21)\) (22 pairs)
- Continue this process until \(x\) can no longer satisfy the equation with a non-negative \(y\).
5. **Practical Limit:**
The maximum value for \(x\) under the integer constraint can be calculated from \(21x \leq 300\):
\[
x \leq \frac{300}{21} \approx 14
\]
Therefore \(x\) can be from \(0\) to \(14\) giving us valid pairs to work with further.
6. **List of Valid Integer Coordinates:**
- The (x, y) integer pairs that satisfy the inequality, while checking from \(x=0\) to \(x=14\) while \(y\) remains a non-negative integer, can be calculated similarly.
This gives you a system on which you can plot points meeting the inequality condition. All resulting points within the feasible region below the line \(21x + 11y = 300\) are valid solutions.
1. **Simplify the Inequality:**
\[
10.5x + 5.5y + 150 \leq 300
\]
Subtract 150 from both sides:
\[
10.5x + 5.5y \leq 150
\]
2. **Clear Decimals:**
To eliminate the decimals, multiply the entire inequality by 2:
\[
21x + 11y \leq 300
\]
3. **Find the Boundary Line:**
The equality corresponding to our inequality is:
\[
21x + 11y = 300
\]
We will find the x- and y-intercepts to help visualize the constraint.
- **To find the x-intercept** (set \(y = 0\)):
\[
21x = 300 \implies x = \frac{300}{21} \approx 14.29 \quad (\text{closest integer } x = 14)
\]
- **To find the y-intercept** (set \(x = 0\)):
\[
11y = 300 \implies y = \frac{300}{11} \approx 27.27 \quad (\text{closest integer } y = 27)
\]
Therefore, the intercepts are approximately:
- \(x\)-intercept at \( (14, 0) \)
- \(y\)-intercept at \( (0, 27) \)
4. **Determine Integer Coordinates:**
We can substitute integer values in the original equation to find valid \((x, y)\) pairs that satisfy the inequality. Given \(x\) and \(y\) must be non-negative integers and keep the total under 300.
Let's try some integer combinations while keeping \(21x + 11y \leq 300\):
- \(x = 0\):
- \(11y \leq 300 \implies y \leq 27\) → Pairs: \((0, 0), (0, 1), ..., (0, 27)\) (28 pairs)
- \(x = 1\):
- \(21(1) + 11y \leq 300 \implies 11y \leq 279 \implies y \leq 25\) → Pairs: \((1, 0), (1, 1), ..., (1, 25)\) (26 pairs)
- \(x = 2\):
- \(21(2) + 11y \leq 300 \implies 11y \leq 258 \implies y \leq 23\) → Pairs: \((2, 0), (2, 1), ..., (2, 23)\) (24 pairs)
- \(x = 3\):
- \(21(3) + 11y \leq 300 \implies 11y \leq 237 \implies y \leq 21\) → Pairs: \((3, 0), (3, 1), ..., (3, 21)\) (22 pairs)
- Continue this process until \(x\) can no longer satisfy the equation with a non-negative \(y\).
5. **Practical Limit:**
The maximum value for \(x\) under the integer constraint can be calculated from \(21x \leq 300\):
\[
x \leq \frac{300}{21} \approx 14
\]
Therefore \(x\) can be from \(0\) to \(14\) giving us valid pairs to work with further.
6. **List of Valid Integer Coordinates:**
- The (x, y) integer pairs that satisfy the inequality, while checking from \(x=0\) to \(x=14\) while \(y\) remains a non-negative integer, can be calculated similarly.
This gives you a system on which you can plot points meeting the inequality condition. All resulting points within the feasible region below the line \(21x + 11y = 300\) are valid solutions.