Final answer:
Graph the line x + 2y = 4 by plotting intercepts and using a dashed line. Test points like (5,0) and (0,3) are solutions, while (0,1) is not. Analytical techniques for solving such problems are more accurate than graphical techniques.
Step-by-step explanation:
To graph the inequality x + 2y > 4, we first graph the line x + 2y = 4 as if it were an equation. This line is the boundary of the inequality. To do this, we find the intercepts: when x = 0, y = 2 and when y = 0, x = 4. We plot these points and draw a straight line through them. Since the inequality is greater than and not greater than or equal to, we will use a dashed line to indicate that points on the line are not included in the solution set.
Next, we need to determine which side of the line represents the solution set. We can pick a test point not on the line, such as the origin (0,0), and substitute it into the inequality. Substituting gives us 0 + 2(0) > 4, which simplifies to 0 > 4, which is false. So the side of the line that does not include the origin is the solution set. We shade this region to indicate all points (x, y) that satisfy the inequality.
Possible solutions to the inequality may be points like (5,0) or (0,3), since substituting these into the inequality gives 5 + 2(0) > 4 which simplifies to 5 > 4 (true) and 0 + 2(3) > 4 which simplifies to 6 > 4 (true), respectively. A non-solution could be the point (0,1) because 0 + 2(1) > 4 simplifies to 2 > 4 (false).
While you can solve this graphically, the analytical technique for solving this problem is potentially more accurate. This is because graphing relies on visual approximations, which can lead to errors, especially when determining the exact region of inequality or the precise slope of the line. Analytical solutions use mathematical operations to clearly define the solution set and are not subject to the same visual inaccuracies.