sistema de inecuaciones 2X2  X+2Y>-1             2X+Y<7

To solve the system of inequalities:

1. \( X + 2Y > -1 \)
2. \( 2X + Y < 7 \)

### Step-by-step approach:

#### 1. Rewrite the inequalities to understand the boundaries:

1. **First inequality:**
   \[
   X + 2Y > -1 \quad \text{(Rewrite this as an equality to plot the boundary: \(X + 2Y = -1\))}
   \]
   - Solve for \( Y \):
     \[
     2Y = -1 - X \quad \Rightarrow \quad Y = \frac{-1 - X}{2}
     \]
   - This line has a slope of \( -\frac{1}{2} \) and a y-intercept of \( -\frac{1}{2} \).
   - The region **above** this line satisfies the inequality.

2. **Second inequality:**
   \[
   2X + Y < 7 \quad \text{(Rewrite this as an equality to plot the boundary: \(2X + Y = 7\))}
   \]
   - Solve for \( Y \):
     \[
     Y = 7 - 2X
     \]
   - This line has a slope of \( -2 \) and a y-intercept of \( 7 \).
   - The region **below** this line satisfies the inequality.

#### 2. Plot the boundary lines and shade the correct regions:

- The first inequality \( X + 2Y > -1 \) corresponds to the region **above** the line \( Y = \frac{-1 - X}{2} \).
- The second inequality \( 2X + Y < 7 \) corresponds to the region **below** the line \( Y = 7 - 2X \).

#### 3. Find the intersection region:
- The solution is the region where both conditions are met: **above** the line \( Y = \frac{-1 - X}{2} \) and **below** the line \( Y = 7 - 2X \).

This region represents all the possible values of \( X \) and \( Y \) that satisfy both inequalities.

Would you like a more detailed graphical explanation or more questions on this?

### 5 Related Questions:
1. How do you graph the boundary lines of an inequality?
2. What is the difference between strict and non-strict inequalities when graphing?
3. How do you find the points of intersection for two inequalities?
4. How do you handle systems of inequalities with more than two variables?
5. What methods can be used to check if a point satisfies a system of inequalities?

**Tip:** Always test a point (like the origin) to verify which side of the boundary line satisfies the inequality.

sistema de inecuaciones 2X2 X+2Y>-1 2X+Y<7

Learn how to solve the system of linear inequalities X+2Y > -1 and 2X+Y < 7 using graphing techniques. This detailed explanation covers how to find boundary lines, the regions that satisfy each inequality, and the intersection of those regions. Suitable for students in Grades 9-11.

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