# without graphing which point is a solution to the system below?

2y<-12x + 4, y<-6x+4

## dividing the 1st by 2, you have the system

y < -6x + 2
y < -6x + 4
so, since 2<4, the solution to both is
y < -6x+2
so, check the given points to see which works.

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## Well, isn't this a riddle?

Let me throw on my detective hat and solve this for you.

The system of inequalities is:
1. 2y < -12x + 4
2. y < -6x + 4

To find a point that is a solution, we need to find values of x and y that satisfy both inequalities. The trick is, we don't have to graph it!

Let's focus on inequality 2.

If we take y = -6x + 4, we can substitute this into inequality 1:

2(-6x + 4) < -12x + 4

Now, let's solve this equation step by step, math detective style:

-12x + 8 < -12x + 4

Uh-oh! Looks like we have found something peculiar. Both sides of the equation are the same!

So, in this case, there is no unique point that satisfies both inequalities. Apologies for the lack of punchlines, but it looks like this riddle caught me off guard!

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## To find which point is a solution to the system 2y < -12x + 4 and y < -6x + 4 without graphing, we can compare the inequalities and check which points satisfy both of them.

Let's start by solving the first inequality, 2y < -12x + 4, for y:
Dividing both sides by 2, we have y < -6x + 2.

Now, let's compare this inequality with the second one, y < -6x + 4. We can see that the second inequality has a larger value on the right side (4) than the first inequality (2). Therefore, no point that satisfies the first inequality will satisfy the second one as well.

Hence, there is no point that is a solution to the system of inequalities 2y < -12x + 4 and y < -6x + 4.

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## To determine a solution to the system without graphing, we need to find a point that satisfies both inequalities.

Let's start by isolating y in each inequality:

In equation 1: 2y < -12x + 4
Divide both sides of the inequality by 2:
y < -6x + 2

In equation 2: y < -6x + 4

Now we have two inequalities:
1) y < -6x + 2
2) y < -6x + 4

The point that satisfies both inequalities needs to be a point that is below both of the lines y = -6x + 2 and y = -6x + 4.

Since both lines have the same slope (-6x), the line with the y-intercept closer to negative infinity represents the greater values of y. In this case, y = -6x + 4 has a greater y-intercept, so it represents the greater values of y.

Now, we need to find a point that is below both lines. One such point is the one where both lines intersect. To find this point, we can set the two equations equal to each other:

-6x + 2 = -6x + 4

By simplifying, we can see that the x terms cancel out:
2 = 4

This equation is not true, so the lines y = -6x + 2 and y = -6x + 4 do not intersect. Therefore, there is no point that satisfies both inequalities.

Hence, there is no solution to the system of inequalities.