https://file.aiquickdraw.com/mathbot/file/172054079440611srsvbh.png

To graph the inequality \(3x - 4y < 4\), we first need to graph the boundary line \(3x - 4y = 4\). Here are the steps:

1. **Rewrite the inequality as an equation**:
   \[
   3x - 4y = 4
   \]

2. **Find the intercepts**:
   - **x-intercept**: Set \(y = 0\)
     \[
     3x - 4(0) = 4 \implies x = \frac{4}{3} \approx 1.33
     \]
   - **y-intercept**: Set \(x = 0\)
     \[
     3(0) - 4y = 4 \implies y = -1
     \]

3. **Plot the intercepts**:
   - Plot the x-intercept at \((1.33, 0)\)
   - Plot the y-intercept at \((0, -1)\)

4. **Draw the boundary line**:
   - Draw a dashed line through the points \((1.33, 0)\) and \((0, -1)\) since the inequality is \(<\) (not \(\leq\)).

5. **Shade the region**:
   - Choose a test point that is not on the line (e.g., the origin \((0,0)\)).
   - Substitute the test point into the inequality \(3x - 4y < 4\):
     \[
     3(0) - 4(0) < 4 \implies 0 < 4 \quad (\text{True})
     \]
   - Since the origin satisfies the inequality, shade the region that includes the origin.

Let's go ahead and graph it accordingly. I'll illustrate it on the provided coordinate plane.

Here is the detailed image description:

- A coordinate plane with x and y axes ranging from -10 to 10.
- The boundary line \(3x - 4y = 4\) is plotted as a dashed line.
- The x-intercept at \((1.33, 0)\) and the y-intercept at \((0, -1)\) are marked.
- The region below the dashed line is shaded, representing \(3x - 4y < 4\).

I'll now generate the image.{
  "prompt": "A coordinate plane with x and y axes ranging from -10 to 10. The boundary line 3x - 4y = 4 is plotted as a dashed line. The x-intercept is marked at (1.33, 0) and the y-intercept at (0, -1). The region below the dashed line is shaded, representing the inequality 3x - 4y < 4.",
  "size": "1024x1024",
  "n": 1
}You've hit your daily maximum number of images. To ensure the best experience for everyone, we have rate limits in place. Please wait for the next day before generating more images. Your daily maximum will reset in 17 hours and 58 minutes.

Here's how you can graph the inequality \(3x - 4y < 4\):

1. **Rewrite the inequality as an equation**:
   \[
   3x - 4y = 4
   \]

2. **Find the intercepts**:
   - **x-intercept**: Set \(y = 0\)
     \[
     3x - 4(0) = 4 \implies x = \frac{4}{3} \approx 1.33
     \]
   - **y-intercept**: Set \(x = 0\)
     \[
     3(0) - 4y = 4 \implies y = -1
     \]

3. **Plot the intercepts**:
   - Plot the x-intercept at \((1.33, 0)\)
   - Plot the y-intercept at \((0, -1)\)

4. **Draw the boundary line**:
   - Draw a dashed line through the points \((1.33, 0)\) and \((0, -1)\) since the inequality is \(<\) (not \(\leq\)).

5. **Shade the region**:
   - Choose a test point that is not on the line (e.g., the origin \((0,0)\)).
   - Substitute the test point into the inequality \(3x - 4y < 4\):
     \[
     3(0) - 4(0) < 4 \implies 0 < 4 \quad (\text{True})
     \]
   - Since the origin satisfies the inequality, shade the region that includes the origin.

Would you like further details or have any other questions?

Here are some questions you might consider asking:
1. How do you solve systems of linear inequalities?
2. What are the steps to graph a linear equation?
3. Can you explain the significance of intercepts in graphing equations?
4. How do you determine which side of the boundary line to shade?
5. What are some applications of linear inequalities in real life?

**Tip:** Always check your boundary line by substituting a point from the shaded region into the original inequality to verify correctness.

Learn how to graph the inequality 3x - 4y < 4 with this detailed guide. Follow the steps to find intercepts, plot the boundary line, and determine the shaded region. Ideal for Grades 7-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation