Worksheets · Precalculus

Systems of inequalities worksheet for Precalculus

Precalculus uses systems of inequalities to describe regions, the setup behind linear programming. The sheet graphs two inequalities and finds the corner where the boundaries meet, writes a system from a graph, and writes one from a word problem about cost and count.

Also taught in Algebra 1 and Algebra 2, each at its own level and with its own examples.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Systems of Inequalities

Date Period

Graph each system, or write it as asked.

  1. Graph and shade the overlap

    Graph y>−2x+2 and y<2x−6−10−10−5−5551010
  2. Graph y>−2x−9 and y≥x−10−10−5−5551010
  3. Graph y≤2x−11 and y≤−2x+5−10−10−5−5551010
  4. Graph y>2x+1 and y≤−x+4−10−10−5−5551010
  5. Graph y≤−2x+6 and y≥x−3−10−10−5−5551010
  6. Graph y<x−4 and y≤−2x−4−10−10−5−5551010
  7. Graph y≤x−1 and y<−2x−7−10−10−5−5551010

  1. Write the system from a graph

    Write the system of inequalities shown by the graph.−10−10−5−5551010
  2. Write the system of inequalities shown by the graph.−10−10−5551010−5
  3. Write the system of inequalities shown by the graph.−10−10−5−5551010
  4. Write the system of inequalities shown by the graph.−10−10−5−5551010
  5. Write the system of inequalities shown by the graph.−10−10−5−5551010
  6. Write the system of inequalities shown by the graph.−10−10−5−5510105
  7. Write the system of inequalities shown by the graph.−10−10−5−5551010

  1. Write a system from a word problem

    You need at least 16 items: x pens at $2 each and y notebooks at $8 each, spending no more than $119. Write the system, including that neither count can be negative.
  2. You need at least 10 items: x bracelets at $3 each and y keychains at $5 each, spending no more than $39. Write the system, including that neither count can be negative.
  3. You need at least 7 items: x cupcakes at $3 each and y cookies at $5 each, spending no more than $33. Write the system, including that neither count can be negative.
  4. You need at least 13 items: x pens at $2 each and y notebooks at $3 each, spending no more than $34. Write the system, including that neither count can be negative.
  5. You need at least 12 items: x cupcakes at $5 each and y cookies at $4 each, spending no more than $56. Write the system, including that neither count can be negative.
  6. You need at least 18 items: x bracelets at $1 each and y keychains at $3 each, spending no more than $47. Write the system, including that neither count can be negative.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Systems of InequalitiesAnswer keyVersion 2

Date Period

Graph each system, or write it as asked.

  1. Graph and shade the overlap

    Graph y>−2x+2 and y<2x−6−10−5−5551010−10the overlap, cornered at (2,−2)
  2. Graph y>−2x−9 and y≥x−10−10−5−5551010the overlap, cornered at (−3,−3)
  3. Graph y≤2x−11 and y≤−2x+5−10−10−5−5510105the overlap, cornered at (4,−3)
  4. Graph y>2x+1 and y≤−x+4−10−10−5−5101055the overlap, cornered at (1,3)
  5. Graph y≤−2x+6 and y≥x−3−10−10−5−5551010the overlap, cornered at (3,0)
  6. Graph y<x−4 and y≤−2x−4−10−10−5551010−5the overlap, cornered at (0,−4)
  7. Graph y≤x−1 and y<−2x−7−10−10−5551010−5the overlap, cornered at (−2,−3)

  1. Write the system from a graph

    Write the system of inequalities shown by the graph.−10−10−5−5551010y≥−12x−1, y>2x+3
  2. Write the system of inequalities shown by the graph.−10−10−5551010−5y≤−2x+2, y≤2x−4
  3. Write the system of inequalities shown by the graph.−10−10−5−5551010y>x+3, y>12x+3
  4. Write the system of inequalities shown by the graph.−10−10−5−5551010y>12x−1, y>−12x−2
  5. Write the system of inequalities shown by the graph.−10−10−5−5551010y≤−2x, y≤x−2
  6. Write the system of inequalities shown by the graph.−10−10−5−5510105y<−x−3, y<12x−3
  7. Write the system of inequalities shown by the graph.−10−10−5−5551010y<−12x+1, y>x−2

  1. Write a system from a word problem

    You need at least 16 items: x pens at $2 each and y notebooks at $8 each, spending no more than $119. Write the system, including that neither count can be negative.x+y≥16, 2x+8y≤119, x≥0, y≥0
  2. You need at least 10 items: x bracelets at $3 each and y keychains at $5 each, spending no more than $39. Write the system, including that neither count can be negative.x+y≥10, 3x+5y≤39, x≥0, y≥0
  3. You need at least 7 items: x cupcakes at $3 each and y cookies at $5 each, spending no more than $33. Write the system, including that neither count can be negative.x+y≥7, 3x+5y≤33, x≥0, y≥0
  4. You need at least 13 items: x pens at $2 each and y notebooks at $3 each, spending no more than $34. Write the system, including that neither count can be negative.x+y≥13, 2x+3y≤34, x≥0, y≥0
  5. You need at least 12 items: x cupcakes at $5 each and y cookies at $4 each, spending no more than $56. Write the system, including that neither count can be negative.x+y≥12, 5x+4y≤56, x≥0, y≥0
  6. You need at least 18 items: x bracelets at $1 each and y keychains at $3 each, spending no more than $47. Write the system, including that neither count can be negative.x+y≥18, x+3y≤47, x≥0, y≥0

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

You need at least 12 items: x mugs at $4 each and y plates at $6 each, spending no more than $60. Write the system.

  1. At least 12 items: x + y ≥ 12.
  2. The cost is 4x + 6y, and it can be at most 60: 4x + 6y ≤ 60.
  3. You cannot buy a negative number of either: x ≥ 0 and y ≥ 0.

The answer is x+y≥12, 4x+6y≤60, x≥0, y≥0.

Where students go wrong

Leaving out the hidden constraints in a word problem. Counts of bracelets or keychains cannot be negative, so x ≥ 0 and y ≥ 0 belong in the system even when the question never states them.

Also called graphing systems of inequalities, feasible region or overlapping half planes.

What you can put on this worksheet

Two inequalities, and the overlap
Graph y≥x+1 and y≥−2x+1 → the overlap, cornered at (0,1)
Write the system from a graph
Write the system of inequalities shown by the graph. → y>−12x, y≤x+1
Write the system from words
You need at least 14 items: x bracelets at $3 each and y keychains at $7 each, spending no more than $69. Write the system, including that neither count can be negative. → x+y≥14, 3x+7y≤69, x≥0, y≥0
Name a point: level lines, standard form, strips
Name one point in the solution:
y≥−1
y≤2x+6 → e.g. (1,5)
Graph two inequalities: more forms
Graph −3x−4y<−15 and 5x−3y≥−4 → Shade where both hold; the lines cross at (1,3)

Questions about these worksheets

Yes. Every systems of inequalities sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the systems of inequalities questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: two inequalities, and the overlap, write the system from a graph, write the system from words, name a point: level lines, standard form, strips and graph two inequalities: more forms. Tick as many as you want and set how many of each, or let it spread them evenly.

Giving the whole shaded area rather than the overlap. Only the part covered by both counts, and it is easy to lose track once two grays sit on top of each other. Shading lightly, or with lines in two directions, keeps it readable.