Systems of Inequalities Worksheets with Answer Key - Free Printable

Since both ExplainAnswer and TeachTopic are false, and GeneratePractice is also false, we will only provide the Final Answer — which in this case means solving each of the six systems of linear inequalities by graphing (as instructed), and describing the solution area for each.

But note: Since we can’t actually draw graphs here, we’ll describe how to graph each inequality and where to shade for each system. The student will then plot these on their own grid.

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Let’s go one by one:

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Problem 1: y > -x - 4 ; 8x - 4y > 1 ; x < 4

Step 1: Graph y = -x - 4 → dashed line (because >), shade above it.

Step 2: Rewrite 8x - 4y > 1 → -4y > -8x + 1 → y < 2x - 0.25 → dashed line, shade below.

Step 3: x < 4 → vertical dashed line at x=4, shade left.

Solution area: Region that satisfies all three — above first line, below second line, left of x=4.

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Problem 2: y ≤ 2x - 3 ; y ≥ -3 ; y ≤ -1.2x + 2.5

Step 1: y = 2x - 3 → solid line, shade below.

Step 2: y = -3 → horizontal solid line, shade above.

Step 3: y = -1.2x + 2.5 → solid line, shade below.

Solution area: Triangle-like region bounded by all three lines — between y=-3 and under both slanted lines.

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Problem 3: y + 3 < 2x ; x ≤ 3

Rewrite: y < 2x - 3 → dashed line, shade below.

x ≤ 3 → vertical solid line at x=3, shade left.

Solution area: Below the line y=2x-3 and left of x=3.

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Problem 4: y < 3 ; y > -x - 3 ; y ≥ 3x - 4

Step 1: y = 3 → dashed horizontal line, shade below.

Step 2: y = -x - 3 → dashed line, shade above.

Step 3: y = 3x - 4 → solid line, shade above.

Solution area: Between the two slanted lines and under y=3.

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Problem 5: x - 2y ≥ 4 ; 2x - y ≥ -2

Rewrite first: x - 2y ≥ 4 → -2y ≥ -x + 4 → y ≤ 0.5x - 2 → solid line, shade below.

Second: 2x - y ≥ -2 → -y ≥ -2x -2 → y ≤ 2x + 2 → solid line, shade below.

Wait — let’s double-check:

Actually, for 2x - y ≥ -2 → subtract 2x: -y ≥ -2x -2 → multiply by -1 (flip inequality): y ≤ 2x + 2 → yes.

So both are “shade below” solid lines.

Find intersection point: solve y = 0.5x - 2 and y = 2x + 2

Set equal: 0.5x - 2 = 2x + 2 → -4 = 1.5x → x = -8/3 ≈ -2.67, y = 2*(-8/3)+2 = -16/3 + 6/3 = -10/3

Solution area: Below both lines — forms a wedge going down-right from intersection.

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Problem 6: x + y > 3 ; y ≤ 2x - 3

First: x + y > 3 → y > -x + 3 → dashed line, shade above.

Second: y = 2x - 3 → solid line, shade below.

Solution area: Above the first line, below the second line — they intersect when -x+3 = 2x-3 → 6 = 3x → x=2, y=1.

So solution is the region starting at (2,1) going up-right between the two lines.

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Now, since the user didn’t ask for explanation or teaching, and we’re to give only final answers — but there’s no single numerical answer — we must interpret “Final Answer” as the completed work: i.e., the descriptions above are what the student should use to graph.

But per instructions: “Provide only the final correct answer — this should come AFTER all explanations and reasoning.”

Since we’ve done the reasoning, and the task is to solve by graphing, and we can’t draw, the best we can do is summarize:

Final Answer:
For each system, graph the boundary lines (dashed for strict inequalities, solid otherwise), shade the correct side of each, and the overlapping shaded region is the solution. Specific regions described above for problems 1–6.