To find an ordered pair that satisfies both inequalities y > 3x + 1 and y < x + 4, we need to analyze what these inequalities represent.
1. **Understanding the inequalities**:
- The first inequality, y > 3x + 1, represents all the points that lie above the line defined by the equation y = 3x + 1.
- The second inequality, y < x + 4, represents all the points that lie below the line defined by the equation y = x + 4.
2. **Finding the lines**:
We can rewrite both inequalities in slope-intercept form:
- y = 3x + 1 has a slope of 3 and a y-intercept of 1.
- y = x + 4 has a slope of 1 and a y-intercept of 4.
3. **Graphing the lines**:
On a graph, you will draw the line y = 3x + 1 as a dashed line (since it’s >, not ≥) and shade above it. For y = x + 4, draw a dashed line and shade below it.
4. **Finding the intersection area**:
The solution will be in the area where the shading overlaps—above the first line and below the second line.
5. **Testing a point**:
To find a specific ordered pair (x, y) that satisfies both, we can test points within the shaded region. For example, let’s test the point (0, 2):
- For the first inequality:
y (2) > 3(0) + 1:
2 > 1 (true). - For the second inequality:
y (2) < 0 + 4:
2 < 4 (true).
Since (0, 2) satisfies both inequalities, we can conclude that the ordered pair (0, 2) makes both inequalities true.
In summary, the ordered pair (0, 2) is one example that satisfies both y > 3x + 1 and y < x + 4.