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

Guest Dec 10, 2017

#1**+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

CPhill
Dec 10, 2017

#2**+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

hectictar
Dec 10, 2017