Question
Graph the inequality on the axes below.
3, x, plus, 5, y, is less than, 40
3x+5y<40
3, x, plus, 5, y, is less than, 40
3x+5y<40
Answers
GPT-4o mini
To graph the inequality \(3x + 5y < 40\), follow these steps:
### Step 1: Convert the inequality into an equation
Start by converting the inequality \(3x + 5y < 40\) into an equation for graphing the boundary line:
\[
3x + 5y = 40
\]
### Step 2: Find the intercepts of the line
To graph the line \(3x + 5y = 40\), find the x-intercept and y-intercept.
- **X-intercept**: Set \(y = 0\) and solve for \(x\):
\[
3x + 5(0) = 40 \implies 3x = 40 \implies x = \frac{40}{3} \approx 13.33
\]
- **Y-intercept**: Set \(x = 0\) and solve for \(y\):
\[
3(0) + 5y = 40 \implies 5y = 40 \implies y = \frac{40}{5} = 8
\]
### Step 3: Plot the line
Now plot the intercepts on a graph. The points to plot are:
- X-intercept: \((\frac{40}{3}, 0) \approx (13.33, 0)\)
- Y-intercept: \((0, 8)\)
Draw a dashed line through these points since the inequality is "less than" and does not include the boundary.
### Step 4: Determine the shading
To determine which side of the line to shade, choose a test point that is not on the line. A common choice is the origin \((0, 0)\).
Plug the test point into the original inequality:
\[
3(0) + 5(0) < 40 \implies 0 < 40
\]
This is true, so shade the side of the line that includes the origin.
### Final Graph:
1. Graph a dashed line through points \((\frac{40}{3}, 0)\) and \((0, 8)\).
2. Shade the area beneath the line (the area that includes the point \((0, 0)\)).
This complete graph represents the solution to the inequality \(3x + 5y < 40\).
### Step 1: Convert the inequality into an equation
Start by converting the inequality \(3x + 5y < 40\) into an equation for graphing the boundary line:
\[
3x + 5y = 40
\]
### Step 2: Find the intercepts of the line
To graph the line \(3x + 5y = 40\), find the x-intercept and y-intercept.
- **X-intercept**: Set \(y = 0\) and solve for \(x\):
\[
3x + 5(0) = 40 \implies 3x = 40 \implies x = \frac{40}{3} \approx 13.33
\]
- **Y-intercept**: Set \(x = 0\) and solve for \(y\):
\[
3(0) + 5y = 40 \implies 5y = 40 \implies y = \frac{40}{5} = 8
\]
### Step 3: Plot the line
Now plot the intercepts on a graph. The points to plot are:
- X-intercept: \((\frac{40}{3}, 0) \approx (13.33, 0)\)
- Y-intercept: \((0, 8)\)
Draw a dashed line through these points since the inequality is "less than" and does not include the boundary.
### Step 4: Determine the shading
To determine which side of the line to shade, choose a test point that is not on the line. A common choice is the origin \((0, 0)\).
Plug the test point into the original inequality:
\[
3(0) + 5(0) < 40 \implies 0 < 40
\]
This is true, so shade the side of the line that includes the origin.
### Final Graph:
1. Graph a dashed line through points \((\frac{40}{3}, 0)\) and \((0, 8)\).
2. Shade the area beneath the line (the area that includes the point \((0, 0)\)).
This complete graph represents the solution to the inequality \(3x + 5y < 40\).
give me a clear explanation of what to graph
GPT-4o mini
Certainly! Let's summarize the steps clearly for graphing the inequality \(3x + 5y < 40\):
### Step-by-Step Instructions:
1. **Convert the Inequality to an Equation**:
- Start with the inequality: \(3x + 5y < 40\).
- For the boundary line, rewrite it as an equation:
\[
3x + 5y = 40.
\]
2. **Find the Intercepts**:
- **X-Intercept**: Set \(y = 0\) and solve for \(x\):
\[
3x + 5(0) = 40 \implies 3x = 40 \implies x = \frac{40}{3} \approx 13.33.
\]
Thus, the x-intercept is \((13.33, 0)\).
- **Y-Intercept**: Set \(x = 0\) and solve for \(y\):
\[
3(0) + 5y = 40 \implies 5y = 40 \implies y = 8.
\]
Thus, the y-intercept is \((0, 8)\).
3. **Plot the Boundary Line**:
- Plot the points you found: \((13.33, 0)\) and \((0, 8)\).
- Connect these points with a dashed line. Use a dashed line because the inequality is "less than" (\(<\)), meaning the boundary is not included in the solution.
4. **Determine Which Side to Shade**:
- Choose a test point, such as \((0, 0)\) (the origin).
- Substitute the test point into the inequality:
\[
3(0) + 5(0) < 40 \implies 0 < 40.
\]
This statement is true, so shade the area of the graph that contains the point \((0, 0)\).
### Summary:
- Plot a dashed line through the points \((13.33, 0)\) and \((0, 8)\).
- Shade **below** this line, including the origin \((0, 0)\).
Your final graph should clearly show the dashed line, dividing the coordinate plane, with the region below the line shaded. This shaded area represents all the values of \((x, y)\) that satisfy the inequality \(3x + 5y < 40\).
### Step-by-Step Instructions:
1. **Convert the Inequality to an Equation**:
- Start with the inequality: \(3x + 5y < 40\).
- For the boundary line, rewrite it as an equation:
\[
3x + 5y = 40.
\]
2. **Find the Intercepts**:
- **X-Intercept**: Set \(y = 0\) and solve for \(x\):
\[
3x + 5(0) = 40 \implies 3x = 40 \implies x = \frac{40}{3} \approx 13.33.
\]
Thus, the x-intercept is \((13.33, 0)\).
- **Y-Intercept**: Set \(x = 0\) and solve for \(y\):
\[
3(0) + 5y = 40 \implies 5y = 40 \implies y = 8.
\]
Thus, the y-intercept is \((0, 8)\).
3. **Plot the Boundary Line**:
- Plot the points you found: \((13.33, 0)\) and \((0, 8)\).
- Connect these points with a dashed line. Use a dashed line because the inequality is "less than" (\(<\)), meaning the boundary is not included in the solution.
4. **Determine Which Side to Shade**:
- Choose a test point, such as \((0, 0)\) (the origin).
- Substitute the test point into the inequality:
\[
3(0) + 5(0) < 40 \implies 0 < 40.
\]
This statement is true, so shade the area of the graph that contains the point \((0, 0)\).
### Summary:
- Plot a dashed line through the points \((13.33, 0)\) and \((0, 8)\).
- Shade **below** this line, including the origin \((0, 0)\).
Your final graph should clearly show the dashed line, dividing the coordinate plane, with the region below the line shaded. This shaded area represents all the values of \((x, y)\) that satisfy the inequality \(3x + 5y < 40\).