How do I solve:

6x+4y>=80
6x+4y<=170

no solutions, or empty set

solve both equations for y:
y >= -1.5x + 20
y <= -1.5x + 42.5
Same slope with different y-intercepts indicates parallel lines;
Graph both, which shows the solution set: solutions for 1st equation are above the line, and solutions for 2nd equation are below the line, which means no point(s) of commonality.

To solve the system of inequalities:

6x + 4y >= 80
6x + 4y <= 170

We can start by graphing the boundary lines of each inequality. Let's first graph the line for the first inequality, 6x + 4y = 80:

Step 1: Rewrite the equation in slope-intercept form (y = mx + b), where "m" is the slope and "b" is the y-intercept.
Start with 6x + 4y = 80:

Subtract 6x from both sides:
4y = -6x + 80

Divide everything by 4:
y = (-6/4)x + 20
Simplify further:
y = (-3/2)x + 20

Step 2: Plot the y-intercept (0, 20) on a coordinate plane, and use the slope to find the next point. The slope is -3/2, which means you go down 3 units and right 2 units from the y-intercept to find another point. Connect the two points to draw the line.

Now let's graph the line for the second inequality, 6x + 4y = 170:

Step 1: Rewrite the equation in slope-intercept form (y = mx + b) using the same steps as before.
6x + 4y = 170:

Subtract 6x from both sides:
4y = -6x + 170

Divide everything by 4:
y = (-6/4)x + 42.5
Simplify further:
y = (-3/2)x + 42.5

Step 2: Plot the y-intercept (0, 42.5) on the same coordinate plane as before, and use the slope to find the next point. Again, the slope is -3/2, so you go down 3 units and right 2 units from the y-intercept to find another point. Connect the two points to draw the line.

Now that we have both lines graphed, we can determine the shaded region that satisfies both inequalities. The shaded area will be the region where the lines overlap or intersect.

In this case, the inequality signs are inclusive (greater than or equal to, and less than or equal to), which means the lines also represent the boundary of the solution set. Shade the region between and including the two lines.

The solution to the system of inequalities is the set of points within or on the shaded region.

Note: If you need to give an exact solution (x, y), you can choose any point within the shaded region and plug its coordinates into the original inequalities to verify that they hold true.