Question

20 points Choose the system of inequalities that best matches the graph below​

Answer 1

Answer:

D

Step-by-step explanation:

To find the system of inequalities, we simply need to find the equation of each line.

So, let’s find the inequality that is represented by each line.

Red Line:

First, let’s determine the slope. To do so, we can use the slope formula:

$m=\frac{y_2-y_1}{x_2-x_1}$

We can use any two points. Let’s use (0, 1) and (1, 3). Substituting them into the slope formula yields:

$m=\frac{3-1}{1-0}=\frac{2}{1}=2$

Therefore, the slope of the red line is 2.

Now, let’s figure out the rest of the equation. We can use the slope-intercept form:

$y=mx+b$

Notice that our y-intercept of our red line is (0, 1). Therefore, substitute 1 for b and 2 for m. This yields:

$y=2x+1$

Now, we need to determine our sign. Notice that the shaded region is below the red line. Therefore, y is less than our equation.

And since our line is shaded, our sign will be less than or equal to. Therefore:

$y\leq 2x+1$

Blue Line:

Again, let’s first find the slope. We can use the two points (0, 2) and (2, 3). Thus:

$m=\frac{3-2}{2-0}=\frac{1}{2}$

So, the slope is 1/2.

We can now use the slope-intercept form. Notice that our y-intercept is (0, 2). So, substitute 1/2 for m and 2 for b. This yields:

$y=\frac{1}{2}x+2$

Finally, we will determine our sign. Since the shaded region is above the blue line, our y is greater than our equation. And since it’s shaded, our sign is greater than or equal to. Therefore, our equation is:

$y\geq\frac{1}{2}x+2$

Thus, all together, our inequalities are:

$y\leq 2x+1\\y\geq \frac{1}{2}x+2$

Therefore, our answer is D