Question
give me a problem similiar to graph >more than or equal to 2x-3
Answers
Answer
give me a lesson on how to answer it
Answered by
GPT-4o mini
Sure! Here’s a similar problem for you:
**Problem:** Graph the inequality \( y < -\frac{1}{2}x + 4 \).
To solve this problem, follow these steps:
1. **Rewrite the inequality** in slope-intercept form (if necessary).
2. **Graph the boundary line** \( y = -\frac{1}{2}x + 4 \) as a dashed line since the inequality is "less than" (not inclusive).
3. **Shade the region** below the line to indicate the area where the inequality holds true.
Would you like help with the solution steps or a different problem?
**Problem:** Graph the inequality \( y < -\frac{1}{2}x + 4 \).
To solve this problem, follow these steps:
1. **Rewrite the inequality** in slope-intercept form (if necessary).
2. **Graph the boundary line** \( y = -\frac{1}{2}x + 4 \) as a dashed line since the inequality is "less than" (not inclusive).
3. **Shade the region** below the line to indicate the area where the inequality holds true.
Would you like help with the solution steps or a different problem?
Answered by
GPT-4o mini
Absolutely! Here's a step-by-step lesson on how to solve and graph the inequality \( y < -\frac{1}{2}x + 4 \).
### Step 1: Understand the Components of the Inequality
The inequality \( y < -\frac{1}{2}x + 4 \) has the following components:
- **Slope**: The coefficient of \( x \) is \(-\frac{1}{2}\), which means for every 1 unit increase in \( x \), \( y \) decreases by \( \frac{1}{2} \).
- **Y-intercept**: The constant term is \( 4 \), which means the line intersects the y-axis at the point \( (0, 4) \).
### Step 2: Graph the Boundary Line
1. **Find the y-intercept**:
- Plot the point \( (0, 4) \) on the graph.
2. **Use the slope to find a second point**:
- Since the slope is \( -\frac{1}{2} \), from \( (0, 4) \), move down 1 unit and right 2 units to find another point. This gives you the point \( (2, 3) \).
3. **Draw the line**:
- Connect the two points \( (0, 4) \) and \( (2, 3) \).
- Since the inequality is \( < \) (not \(\leq\)), draw a **dashed line** to indicate that the points on the line are not included in the solution set.
### Step 3: Determine Which Side to Shade
1. **Choose a test point**:
- A common test point is the origin \( (0, 0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality:
\[
0 < -\frac{1}{2}(0) + 4
\]
Simplifying gives:
\[
0 < 4 \, (\text{true})
\]
- Since this statement is true, we shade the area that includes the point \( (0, 0) \), which is **below the line**.
### Step 4: Finalize the Graph
- Make sure to clearly indicate the dashed line and the shaded region. The shaded area represents all the points \( (x,y) \) that satisfy the inequality \( y < -\frac{1}{2}x + 4 \).
### Summary
1. Identify the slope and y-intercept.
2. Graph the boundary line using a dashed line for \( < \).
3. Use a test point to determine which side of the line to shade.
4. Shade the correct region.
### Practice Problems
1. Graph the inequality \( y \geq 3x - 1 \).
2. Graph the inequality \( y \leq -x + 2 \).
3. Graph the inequality \( y > \frac{2}{3}x + 1 \).
Would you like further clarification on any of the steps or assistance with the practice problems?
### Step 1: Understand the Components of the Inequality
The inequality \( y < -\frac{1}{2}x + 4 \) has the following components:
- **Slope**: The coefficient of \( x \) is \(-\frac{1}{2}\), which means for every 1 unit increase in \( x \), \( y \) decreases by \( \frac{1}{2} \).
- **Y-intercept**: The constant term is \( 4 \), which means the line intersects the y-axis at the point \( (0, 4) \).
### Step 2: Graph the Boundary Line
1. **Find the y-intercept**:
- Plot the point \( (0, 4) \) on the graph.
2. **Use the slope to find a second point**:
- Since the slope is \( -\frac{1}{2} \), from \( (0, 4) \), move down 1 unit and right 2 units to find another point. This gives you the point \( (2, 3) \).
3. **Draw the line**:
- Connect the two points \( (0, 4) \) and \( (2, 3) \).
- Since the inequality is \( < \) (not \(\leq\)), draw a **dashed line** to indicate that the points on the line are not included in the solution set.
### Step 3: Determine Which Side to Shade
1. **Choose a test point**:
- A common test point is the origin \( (0, 0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality:
\[
0 < -\frac{1}{2}(0) + 4
\]
Simplifying gives:
\[
0 < 4 \, (\text{true})
\]
- Since this statement is true, we shade the area that includes the point \( (0, 0) \), which is **below the line**.
### Step 4: Finalize the Graph
- Make sure to clearly indicate the dashed line and the shaded region. The shaded area represents all the points \( (x,y) \) that satisfy the inequality \( y < -\frac{1}{2}x + 4 \).
### Summary
1. Identify the slope and y-intercept.
2. Graph the boundary line using a dashed line for \( < \).
3. Use a test point to determine which side of the line to shade.
4. Shade the correct region.
### Practice Problems
1. Graph the inequality \( y \geq 3x - 1 \).
2. Graph the inequality \( y \leq -x + 2 \).
3. Graph the inequality \( y > \frac{2}{3}x + 1 \).
Would you like further clarification on any of the steps or assistance with the practice problems?