Question
Which is the graph of the solution set of the inequality 3x−4y≤24
?
HINT: What is a Solution Set to system of inequalities and Linear Inequalties
(1 point)
Responses
?
HINT: What is a Solution Set to system of inequalities and Linear Inequalties
(1 point)
Responses
Answers
GPT-4o mini
To graph the solution set for the inequality \(3x - 4y \leq 24\), you can follow these steps:
1. **Convert the Inequality to Equality**: Start by rewriting the inequality as an equation:
\[
3x - 4y = 24
\]
2. **Find Intercepts**: To graph the line, find the intercepts.
- **X-Intercept**: Set \(y = 0\) and solve for \(x\):
\[
3x - 4(0) = 24 \implies 3x = 24 \implies x = 8
\]
- **Y-Intercept**: Set \(x = 0\) and solve for \(y\):
\[
3(0) - 4y = 24 \implies -4y = 24 \implies y = -6
\]
3. **Plot the Line**: Now plot the points \((8, 0)\) and \((0, -6)\) on a graph and draw the line through them.
4. **Determine the Region**: The inequality \(3x - 4y \leq 24\) indicates that we want the region below (or on) the line. Since the inequality is \(\leq\), the line itself will be included in the solution set, which means it will be a solid line.
5. **Choose a Test Point**: You can test a point not on the line, such as \((0, 0)\):
\[
3(0) - 4(0) \leq 24 \implies 0 \leq 24 \text{ (true)}
\]
Therefore, the region that includes \((0, 0)\) is part of the solution set.
6. **Shading the Region**: Shade the region below the line to represent all the solutions to the inequality \(3x - 4y \leq 24\).
In summary, the graph of the solution set consists of the line \(3x - 4y = 24\) as a solid line and the region below it shaded to indicate all the points \((x, y)\) that satisfy the inequality.
1. **Convert the Inequality to Equality**: Start by rewriting the inequality as an equation:
\[
3x - 4y = 24
\]
2. **Find Intercepts**: To graph the line, find the intercepts.
- **X-Intercept**: Set \(y = 0\) and solve for \(x\):
\[
3x - 4(0) = 24 \implies 3x = 24 \implies x = 8
\]
- **Y-Intercept**: Set \(x = 0\) and solve for \(y\):
\[
3(0) - 4y = 24 \implies -4y = 24 \implies y = -6
\]
3. **Plot the Line**: Now plot the points \((8, 0)\) and \((0, -6)\) on a graph and draw the line through them.
4. **Determine the Region**: The inequality \(3x - 4y \leq 24\) indicates that we want the region below (or on) the line. Since the inequality is \(\leq\), the line itself will be included in the solution set, which means it will be a solid line.
5. **Choose a Test Point**: You can test a point not on the line, such as \((0, 0)\):
\[
3(0) - 4(0) \leq 24 \implies 0 \leq 24 \text{ (true)}
\]
Therefore, the region that includes \((0, 0)\) is part of the solution set.
6. **Shading the Region**: Shade the region below the line to represent all the solutions to the inequality \(3x - 4y \leq 24\).
In summary, the graph of the solution set consists of the line \(3x - 4y = 24\) as a solid line and the region below it shaded to indicate all the points \((x, y)\) that satisfy the inequality.