Question
Describe the error
X+3y<6
X+3y<6
Answers
GPT-4o mini
The expression \(X + 3y < 6\) represents a linear inequality in two variables, \(X\) (or \(x\)) and \(y\). However, the equation appears to contain an error in terms of variable notation.
1. **Variable Naming:** Typically, variables are denoted using lower-case letters. So, instead of using uppercase \(X\), it is conventional to use lowercase \(x\). The correct form would be \(x + 3y < 6\).
2. **Interpreting the Inequality:** This inequality describes a half-plane in the coordinate system where the sum of \(x\) and three times \(y\) is less than 6.
To analyze the inequality further:
- You can rewrite the inequality in a more standard form by isolating \(y\):
\[
3y < 6 - x \quad \Rightarrow \quad y < \frac{6 - x}{3}
\]
This shows that for any given value of \(x\), \(y\) must be less than the result of \(\frac{6 - x}{3}\).
3. **Graphing:** The inequality can be graphed as follows:
- First, graph the line \(x + 3y = 6\) by finding the intercepts:
- For \(x = 0\): \(3y = 6 \Rightarrow y = 2\) (y-intercept).
- For \(y = 0\): \(x = 6\) (x-intercept).
- Then, shade below the line since it's a "less than" inequality.
In summary, the error lies mainly in the use of a capital letter for the variable, and the context is that it's a linear inequality that can be analyzed and graphed accordingly.
1. **Variable Naming:** Typically, variables are denoted using lower-case letters. So, instead of using uppercase \(X\), it is conventional to use lowercase \(x\). The correct form would be \(x + 3y < 6\).
2. **Interpreting the Inequality:** This inequality describes a half-plane in the coordinate system where the sum of \(x\) and three times \(y\) is less than 6.
To analyze the inequality further:
- You can rewrite the inequality in a more standard form by isolating \(y\):
\[
3y < 6 - x \quad \Rightarrow \quad y < \frac{6 - x}{3}
\]
This shows that for any given value of \(x\), \(y\) must be less than the result of \(\frac{6 - x}{3}\).
3. **Graphing:** The inequality can be graphed as follows:
- First, graph the line \(x + 3y = 6\) by finding the intercepts:
- For \(x = 0\): \(3y = 6 \Rightarrow y = 2\) (y-intercept).
- For \(y = 0\): \(x = 6\) (x-intercept).
- Then, shade below the line since it's a "less than" inequality.
In summary, the error lies mainly in the use of a capital letter for the variable, and the context is that it's a linear inequality that can be analyzed and graphed accordingly.