Find a linear inequality with the following solution set. Each grid line represents one unit.
Answer in standard notation
1. https://latex.artofproblemsolving.com/f/b/3/fb366296576fe86f9f1c3bf32bf77f860a26dd72.png
2.https://latex.artofproblemsolving.com/5/e/6/5e65ff3ab12a357e8cd5a5a29228a65b2e0e5fd7.png
For the first.....the points (1, -2) and (3,1) are on the graph
The slope between these is [ 1 - - 2 ] / [ 3 - 1 ] = 3/2
And the equation of the line connecting these points is
y = (3/2)(x -1) - 2
y = (3/2)x -3/2 - 2
y = (3/2)x - 7/2
Multiply through by 2
2y = 3x - 7 rearrange as
3x - 2y = 7
Since we have a dashed line...we are going to have a "<" or ">" sign involved...put a point in the "yelllow" into this equation to see which inequality sign we need.....(0,0) seems good
3(0) - 2(0) = 7 ???
0 = 7 ???
And its clear that we need the "<" sign
So.....the equation is
3x - 2y < 7
Here's the graph : https://www.desmos.com/calculator/vn7sb63doc
Here's the second one:
The red line passes through the points (0, 1) and (1, 0) , so
its slope = [ 0 - 1 ] / [ 1 - 0 ] = -1
Using the point (0, 1) and the slope -1 , the equation of the red line is…
y - 1 = -1x Add 1 to both sides of the equation.
y = -x + 1 Add x to both sides of the equation.
x + y = 1
We want the solutions to include all points to one side of the line, so the inequality will be either
x + y ≤ 1 or x + y ≥ 1
The point (0, 0) is in the yellow region, so we want (0, 0) to make the inequality true.
0 + 0 ≤ 1 or 0 + 0 ≥ 1
0 ≤ 1 true or 0 ≥ 1 false
So the inequality that has (0, 0) as a solution is the one we want, which is…
x + y ≤ 1