Linear inequalities
Looking at two linear inequalities, 5𝑥−2𝑦<10 and 𝑦≤𝑥, how can you identify the x-intercept of boundary and y-intercept of boundary for equation (1). Also, identify the graph of inequality equation ( 1) as a solid line or a dashed line. Lastly, provide the procedural steps how to graph these inequalities.
Your initial response should be 100-200 words in length
Discussion Board Reply
Reply from K. Gray
To find the x-intercept of the boundary for the equation 5x – 2y < 10, we set y = 0 and solve for x:
5x – 2(0) = 10
5x = 10
x = 2
So, the x-intercept is at (2, 0).
To find the y-intercept of the boundary, we set x = 0 and solve for y:
5(0) – 2y = 10
-2y = 10
y = -5
Therefore, the y-intercept is at (0, -5).
The graph of the inequality 5x – 2y < 10 will have a dashed line because the inequality does not include the equal sign.
To graph these inequalities:
- Start by graphing the line 5x – 2y = 10. To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y. Plot these points and draw a dashed line through them.
- Next, graph the line y = x. This line is a solid line because the inequality includes the equal sign.
- Choose a test point not on the line. For example, (0,0) is a common choice.
- Substitute the test point into each inequality. If it’s true, shade the region that contains the test point. If it’s false, shade the other region.
- The solution to the system of inequalities is the overlapping shaded region.
Sample Answer
Great response, K. Gray!
Your explanation is clear and well-organized. You accurately identified the intercepts, shading instruction, and the dashed line for the first inequality.