Graph the solution for the following system of inequalities. $$ x+y3 $$

Question

Graph the solution for the following system of inequalities. $$ x+y>3 $$

UpStudy AI · Accepted Answer

To graph the inequality $$ x + y > 3 $$, follow these steps:

### 1. **Graph the Boundary Line**

First, consider the equation $$ x + y = 3 $$. This line serves as the boundary for the inequality.

- **Find the intercepts:**
  - **When $$ x = 0 $$:**
    $$
    0 + y = 3 \implies y = 3
    $$
    So, the point $$(0, 3)$$ is on the line.
  
  - **When $$ y = 0 $$:**
    $$
    x + 0 = 3 \implies x = 3
    $$
    So, the point $$(3, 0)$$ is on the line.

- **Plot the Line:**
  - Draw a straight line passing through the points $$(0, 3)$$ and $$(3, 0)$$.
  - Since the inequality is $$>$$ (strictly greater than), use a **dashed line** to indicate that points on the line $$ x + y = 3 $$ are **not** included in the solution.

![Graph of x + y > 3](https://i.imgur.com/7BvZUAA.png)

### 2. **Determine the Shading Direction**

The inequality $$ x + y > 3 $$ indicates that we want all the points where the sum of $$ x $$ and $$ y $$ is greater than 3. To figure out which side of the line to shade:

- **Test a Point Not on the Line:**
  - A common test point is the origin $$(0, 0)$$.
  - Substitute into the inequality:
    $$
    0 + 0 > 3 \implies 0 > 3 \quad \text{(False)}
    $$
  - Since the origin does **not** satisfy the inequality, the region **not** containing the origin is the solution set.

- **Shade the Appropriate Region:**
  - Shade the area **above** the dashed line $$ x + y = 3 $$. This represents all points where $$ x + y > 3 $$.

### 3. **Final Graph**

Here's a visual representation of the graph:

```
y
|
|          * (0,3)
|         /
|        /
|       /
|      /
|     / <--- Shaded region is above this dashed line
|    /
|   /
|  / 
| / 
|/____________ x
       (3,0)
```

- The dashed line represents $$ x + y = 3 $$.
- The shaded region above the line is where $$ x + y > 3 $$.

### 4. **Alternative Method: Solving for y**

You can also rewrite the inequality to solve for $$ y $$ and make graphing easier:

$$
x + y > 3 \implies y > -x + 3
$$

- **Graph the Line $$ y = -x + 3 $$:**
  - This is the same boundary line as $$ x + y = 3 $$, with a slope of $$-1$$ and a y-intercept at $$ (0, 3) $$.
  - Use a **dashed line** since the inequality is strict.

- **Shade Above the Line:**
  - $$ y > -x + 3 $$ means shading above the line.

### 5. **Summary**

- **Boundary Line:** Graph $$ x + y = 3 $$ as a dashed line.
- **Shading:** Shade the region **above** the dashed line.
- **Solution Set:** All points $$(x, y)$$ where the sum of $$ x $$ and $$ y $$ is greater than 3.

Feel free to use graphing tools or software like Desmos for a visual representation:

[Desmos Graphing Calculator](https://www.desmos.com/calculator)

---

If you have any further questions or need additional assistance with graphing inequalities, feel free to ask!
To graph the inequality $$ x + y > 3 $$: 1. **Draw the boundary line** $$ x + y = 3 $$ as a dashed line. 2. **Shade the region above** the dashed line. 3. **Test a point** (e.g., the origin $$(0,0)$$) to confirm the shading direction. The shaded area represents all points where $$ x + y > 3 $$.

Graph the solution for the following system of inequalities. \( x+y>3 \)

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