Question
Which is the graph of the solution set of the inequality 3x−4y≤24
?(1 point)
Responses
?(1 point)
Responses
Answers
GPT-4o mini
To graph the solution set of the inequality \(3x - 4y \leq 24\), follow these steps:
1. **Rewrite the inequality in slope-intercept form**:
Start by isolating \(y\):
\[
-4y \leq -3x + 24
\]
Divide each term by -4 (remember to flip the inequality sign):
\[
y \geq \frac{3}{4}x - 6
\]
2. **Graph the boundary line**:
The boundary line of the inequality is given by \(y = \frac{3}{4}x - 6\). To graph this line:
- Identify the y-intercept: When \(x = 0\), \(y = -6\).
- Identify another point. For example, when \(x = 4\):
\[
y = \frac{3}{4}(4) - 6 = 3 - 6 = -3
\]
- Plot the points (0, -6) and (4, -3) on a graph and draw the line. Since the inequality is \(\leq\), this line will be solid.
3. **Determine the shaded region**:
The inequality \(y \geq \frac{3}{4}x - 6\) means that you will shade the area above (or on) the line.
4. **Testing a point**:
To confirm, you can test a point not on the line, such as (0, 0):
\[
3(0) - 4(0) \leq 24 \implies 0 \leq 24 \quad \text{(true)}
\]
Thus, (0, 0) is in the solution set, and the region above the line is correct.
So, the graph of the inequality \(3x - 4y \leq 24\) shows a solid line passing through points like (0, -6) and (4, -3), with the area above this line shaded.
1. **Rewrite the inequality in slope-intercept form**:
Start by isolating \(y\):
\[
-4y \leq -3x + 24
\]
Divide each term by -4 (remember to flip the inequality sign):
\[
y \geq \frac{3}{4}x - 6
\]
2. **Graph the boundary line**:
The boundary line of the inequality is given by \(y = \frac{3}{4}x - 6\). To graph this line:
- Identify the y-intercept: When \(x = 0\), \(y = -6\).
- Identify another point. For example, when \(x = 4\):
\[
y = \frac{3}{4}(4) - 6 = 3 - 6 = -3
\]
- Plot the points (0, -6) and (4, -3) on a graph and draw the line. Since the inequality is \(\leq\), this line will be solid.
3. **Determine the shaded region**:
The inequality \(y \geq \frac{3}{4}x - 6\) means that you will shade the area above (or on) the line.
4. **Testing a point**:
To confirm, you can test a point not on the line, such as (0, 0):
\[
3(0) - 4(0) \leq 24 \implies 0 \leq 24 \quad \text{(true)}
\]
Thus, (0, 0) is in the solution set, and the region above the line is correct.
So, the graph of the inequality \(3x - 4y \leq 24\) shows a solid line passing through points like (0, -6) and (4, -3), with the area above this line shaded.