Question

Solve the system of inequalities by graphing.
[1]: y≤–7x+3
[2]: y≤–1/6x+7
Select a line to change it between solid and dotted. Select a region to shade it.
describe exact points to plot for both equations.

Answer 1

Steps for graphing and the exact points to plot

1) Line type
- Both inequalities are "≤", so draw both boundary lines as solid (not dotted). Points on the lines are included.

2) Points to plot for y = -7x + 3
- y-intercept: (0, 3).
- Another integer point: (1, -4) because y = -7(1) + 3 = -4.
- Optional x-intercept (exact): (3/7, 0).

3) Points to plot for y = -1/6 x + 7
- y-intercept: (0, 7).
- Another integer point: (6, 6) because y = -1/6·6 + 7 = -1 + 7 = 6.
- Optional x-intercept (exact): (42, 0).

4) Intersection of the two lines (exact)
- Solve -7x + 3 = -1/6 x + 7:
Multiply by 6: -42x + 18 = -x + 42 → -41x = 24 → x = -24/41.
y = -7x + 3 = 291/41.
- Intersection point: (-24/41, 291/41).

5) Region to shade
- Shade the region that satisfies both inequalities: y ≤ -7x + 3 and y ≤ -1/6 x + 7. That is the overlap of the half-planes below (and including) each line.
- A quick test point: (0,0) satisfies 0 ≤ 3 and 0 ≤ 7, so (0,0) is in the solution. Shade the region that contains the origin.

Final description to draw on your graph
- Plot and connect the points (0,3) and (1,-4) with a solid line for y = -7x + 3.
- Plot and connect the points (0,7) and (6,6) with a solid line for y = -1/6 x + 7.
- Mark the intersection at (-24/41, 291/41).
- Shade the region that is below both solid lines (the overlapping region, including the boundary lines).