What is a solution to the simultaneous inequalities y > 3x + 1 and y ≤ 2x + 5? Write the x-value first and the y-value second.
To find the solution to the simultaneous inequalities, we need to first graph the two lines represented by the inequalities.
For the inequality y > 3x + 1, we can plot the points (0, 1) and (1, 4) and draw a line passing through them. The line will go upwards, with a slope of 3.
For the inequality y ≤ 2x + 5, we can plot the points (0, 5) and (1, 7) and draw a line passing through them. The line will go upwards, but with a smaller slope of 2.
Next, we need to shade the area that satisfies both inequalities. Since the area above the line y > 3x + 1 and below the line y ≤ 2x + 5 satisfies both, we shade that area.
The solution to the simultaneous inequalities is the region shaded between the two lines.
Looking at the graph, we can see that the x-value first increases. So, the x-values increase from left to right on the graph.
As for the y-values, we can see that they generally increase as well. However, we can note that as we move from left to right on the graph, the values satisfying y > 3x + 1 increase faster than the values satisfying y ≤ 2x + 5.
Therefore, we can list the x-values and y-values that satisfy both inequalities in increasing order:
(1, 4), (2, 5), (3, 6), (4, 7), (5, y > 16), ...
So, the solution to the simultaneous inequalities, with the x-value first and the y-value second, is:
(1, 4), (2, 5), (3, 6), (4, 7), (5, y > 16), ...
Note: Since the inequalities are not strict (i.e., they include the equal sign), we include points on the lines as solutions as well.