Solving Systems of Inequalities worksheet with four graphed examples.

Let’s solve each system of inequalities step by step. We’ll graph each inequality and find where the shaded regions overlap — that’s the solution.

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Problem 1:
> y ≤ -x - 2
> y ≥ -5x + 2

Step 1: Graph y = -x - 2
- Slope = -1, y-intercept = -2 → plot (0, -2), then go down 1, right 1 to (1, -3)
- Since it’s “≤”, draw a solid line and shade below

Step 2: Graph y = -5x + 2
- Slope = -5, y-intercept = 2 → plot (0, 2), then go down 5, right 1 to (1, -3)
- Since it’s “≥”, draw a solid line and shade above

Step 3: Find overlapping region
- The two lines intersect at (-1, -1) [you can check: plug x=-1 into both equations]
- Shade below first line AND above second line → small triangle near bottom left

✔ Final Answer for #1: The solution is the region bounded between the two solid lines, including the intersection point (-1, -1), shaded in the lower-left area.

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Problem 2:
> y > -x - 2
> y < -5x + 2

Step 1: Graph y = -x - 2
- Same as before, but now “>” → dashed line, shade above

Step 2: Graph y = -5x + 2
- Same line, but “<” → dashed line, shade below

Step 3: Overlap?
- Above first dashed line AND below second dashed line → this creates a narrow strip between them, NOT including the lines themselves
- Intersection still at (-1, -1), but since both are strict inequalities, that point is NOT included

✔ Final Answer for #2: The solution is the open region between the two dashed lines, not including any points on the lines.

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Problem 3:
> y ≤ (1/2)x + 2
> y < -2x - 3

Step 1: Graph y = (1/2)x + 2
- Slope = 1/2, y-intercept = 2 → plot (0, 2), then up 1, right 2 to (2, 3)
- “≤” → solid line, shade below

Step 2: Graph y = -2x - 3
- Slope = -2, y-intercept = -3 → plot (0, -3), then down 2, right 1 to (1, -5)
- “<” → dashed line, shade below

Step 3: Overlap?
- Below both lines → look for region under both graphs
- They intersect when: (1/2)x + 2 = -2x - 3 → multiply by 2: x + 4 = -4x -6 → 5x = -10 → x = -2 → y = (1/2)(-2)+2 = 1 → so (-2, 1)
- But since second inequality is strict, point (-2,1) is NOT included

✔ Final Answer for #3: The solution is the region below both lines, with the boundary from the first line included (solid), and the second line excluded (dashed). Shaded area is in the lower-left quadrant.

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Problem 4:
> x ≤ -3
> y < (5/3)x + 2

Step 1: Graph x = -3
- Vertical line at x = -3
- “≤” → solid line, shade left of it

Step 2: Graph y = (5/3)x + 2
- Slope = 5/3, y-intercept = 2 → plot (0, 2), then up 5, right 3 to (3, 7)
- “<” → dashed line, shade below

Step 3: Overlap?
- Left of x = -3 AND below the slanted dashed line
- At x = -3, y < (5/3)(-3) + 2 = -5 + 2 = -3 → so below y = -3 at that x-value
- Region is infinite to the left and downward

✔ Final Answer for #4: The solution is all points to the left of or on the vertical line x = -3, AND below the dashed line y = (5/3)x + 2.

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Final Answer:

1) Solution is the closed triangular region bounded by the two solid lines, including the vertex at (-1, -1).

2) Solution is the open region between the two dashed lines, excluding all boundary points.

3) Solution is the region below both lines; includes the solid line y = (1/2)x + 2, excludes the dashed line y = -2x - 3.

4) Solution is the region to the left of or on x = -3, and below the dashed line y = (5/3)x + 2.