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# Help

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Find a linear inequality with the following solution set. Each grid line represents one unit.

1. https://latex.artofproblemsolving.com/f/b/3/fb366296576fe86f9f1c3bf32bf77f860a26dd72.png

2.https://latex.artofproblemsolving.com/5/e/6/5e65ff3ab12a357e8cd5a5a29228a65b2e0e5fd7.png

Dec 10, 2017

#1
+95883
+2

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

Dec 10, 2017
edited by CPhill  Dec 10, 2017
edited by CPhill  Dec 10, 2017
#2
+7348
+1

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

Dec 10, 2017