Graphing systems of inequalities

[bosco]

I'm struggling with the problem below.



I believe the second choice is the answer. The graph of 2x - y > 2 is the dashed line. The slope is -2, the y-intercept is 2, and the line is dashed because it does not contain the "or equal to" line.
The solid line is the graph of x - 3y = 6... the slope is 1/3 and the y-intercept is -2. But here's what has me stumped... If this is supposed to be x - 3y is "less than or equal to" 6, shouldn't the shaded region be BELOW the line rather than above it? Shouldn't the shaded region be the lower right region instead of the upper right region?
Clearly none of the other choices are correct, though. I can rule out answers 1 and 4 because the system of inequalities needs to have one symbol with an "or equal to" sign and one without because of the dashed and solid lines. I can rule out answer 3 because the 2x + y equation has a dashed line and would not have the "or equal to" sign.
But the direction of the inequality symbol in x - 3y is less than or equal to 6 has me thrown off. I'd say the graph indicates answer 2, but only if it were x - 3y is greater than or equal to 6.
Am I missing something??

Thanks for any help you can give.

 

[ksdhart2]

Your intuition is good, leading you to what appears to you as the correct answer. But is it? All that remains is to clear up any confusion surrounding whether the shaded region is above the line o or below it. To do that, let's think about what the inequality really means. We're given:

\(\displaystyle x - 3y \le 6\)

With a bit of algebra, we can see that's equivalent to:

\(\displaystyle x \le 6 + 3y\)

So, now we know that for any x we pick, it will be on the graph so long as it's less than 6 + 3y. The graph is shown from -8 to 8, so let's start at x = -8 and see what we get:

\(\displaystyle -8 \le 6 + 3y\)

Solving this equation reveals:

\(\displaystyle -14/3 \le y\) or \(\displaystyle y \ge -14/3\)

So let's make a new graph. Draw the line x = -8 for y values greater than -14/3. You can stop at y = 6 because that's the only portion the shown graph is concerned with. Is this line segment you just drew above or below the line x - 3y = 6? Why or why not? Let's maybe try another one, say x = -7:

\(\displaystyle -7 \le 6 + 3y\)

\(\displaystyle y \ge -13/3\)

Again, graph the line x = -7 for y values greater than -13/3. Is this line segment you just drew above or below the line x - 3y = 6? Why or why not? Whichever conclusion you came to, can you see why it must be true for any x value you pick? Thus, if you were to draw all of the possible line segments from x = -8 to 8, would you see the graph as shown in the picture? If you do, then you know your answer is correct. If you don't, then you know the problem has an error somewhere and you need to ask your teacher for further clarification.

 

[stapel]

View attachment 8198

I believe the second choice is the answer....

Sometimes it's quicker to just figure out the lines yourself, and then find the answer that matches what you've already gotten:

The line going from the lower left upward to the right has a slope of up-two, over-six, or m = 2/6 = 1/3; it has a y-intercept of b = -2. So its equation is y = (1/3)x - 2. The line is solid, and the shading is above the line, so the inequality is y > (1/3)x - 2. Rearranging into their preferred format:

. . . . .3y > x - 6

. . . . .6 > x - 3y

. . . . .x - 3y < 6

The line coming from the upper left going downward to the right has a slope of down-two, over one, or m = -2/1 = -2; it has a y-intercept at b = 2. So the line equation is y = -2x + 2. The line is dashed and the shading is above the line, so the inequality is y > -2x + 2. Rearranging into their preferred format:

. . . . .-2 > -2x - y

. . . . .2 < 2x + y

. . . . .2x + y > 2

Now it's obvious which answer-option is correct.

 

[bosco]

Thank you. That's very helpful.

 

[bosco]

Now it's obvious which answer-option is correct.

Perfectly clear. Thank you!