Graphing systems of inequalities Worksheets - Free Printable

I'll solve this system of inequalities step by step.

Step 1: Graph the first inequality y > x + 3
- First, I'll graph the line y = x + 3 (dashed line because it's strictly greater than)
- This line has slope 1 and y-intercept 3
- Points on this line: (0, 3), (-3, 0), (1, 4), (-1, 2)
- Since y > x + 3, shade ABOVE the line

Step 2: Graph the second inequality y ≤ -5x - 3
- First, I'll graph the line y = -5x - 3 (solid line because it includes equals)
- This line has slope -5 and y-intercept -3
- Points on this line: (0, -3), (-1, 2), (1, -8)
- Since y ≤ -5x - 3, shade BELOW the line

Step 3: Find the intersection point of the two lines
Set them equal to find where they meet:
x + 3 = -5x - 3
x + 5x = -3 - 3
6x = -6
x = -1

Substitute back: y = (-1) + 3 = 2

So the lines intersect at (-1, 2)

Step 4: Determine the solution region
The solution is where BOTH shaded regions overlap:
- Above y = x + 3 AND below y = -5x - 3
- This creates a wedge-shaped region that opens to the left from the point (-1, 2)

Step 5: Verify with a test point
Let me check point (-2, 0):
- Is 0 > (-2) + 3? → 0 > 1? NO

Let me check point (-2, 2):
- Is 2 > (-2) + 3? → 2 > 1? YES ✓
- Is 2 ≤ -5(-2) - 3? → 2 ≤ 10 - 3? → 2 ≤ 7? YES ✓

So (-2, 2) is in the solution region.

The solution region is the area above the line y = x + 3 and below the line y = -5x - 3, which forms an infinite wedge extending to the left from the intersection point (-1, 2).

Final Answer:
The solution is the region above the dashed line y = x + 3 and below the solid line y = -5x - 3. The boundary lines intersect at (-1, 2), and the solution region extends infinitely to the left from this point.