Graph y ≤ 3x - 12 and y ≥ x - 8

the overlap, cornered at (2, -6)

You need at least 13 items: x pens at $1 each and y notebooks at $4 each, spending no more than $46. Write the system.

x + y ≥ 13, x + 4y ≤ 46, x ≥ 0, y ≥ 0

Graph y ≤ 3x + 17 and y < -x - 7

the overlap, cornered at (-6, -1)

You need at least 8 items: x cupcakes at $2 each and y cookies at $4 each, spending no more than $21. Write the system.

x + y ≥ 8, 2x + 4y ≤ 21, x ≥ 0, y ≥ 0

Graph y ≥ -x - 4 and y < 3x - 8

the overlap, cornered at (1, -5)

You need at least 18 items: x pens at $5 each and y notebooks at $2 each, spending no more than $96. Write the system.

x + y ≥ 18, 5x + 2y ≤ 96, x ≥ 0, y ≥ 0

Graph y < 3x - 15 and y < -3x + 15

the overlap, cornered at (5, 0)

You need at least 13 items: x pens at $1 each and y notebooks at $7 each, spending no more than $83. Write the system.

x + y ≥ 13, x + 7y ≤ 83, x ≥ 0, y ≥ 0

Graph y > -x + 2 and y > 2x + 11

the overlap, cornered at (-3, 5)

You need at least 9 items: x cupcakes at $4 each and y cookies at $3 each, spending no more than $38. Write the system.

x + y ≥ 9, 4x + 3y ≤ 38, x ≥ 0, y ≥ 0

Graph y ≥ -3x - 22 and y ≤ x + 2

the overlap, cornered at (-6, -4)

You need at least 19 items: x cupcakes at $2 each and y cookies at $8 each, spending no more than $143. Write the system.

x + y ≥ 19, 2x + 8y ≤ 143, x ≥ 0, y ≥ 0

the overlap, cornered at (-3, -2)

Graph y ≥ x + 1 and y ≥ -2x + 1

the overlap, cornered at (0, 1)

You need at least 14 items: x bracelets at $3 each and y keychains at $7 each, spending no more than $51. Write the system.

x + y ≥ 14, 3x + 7y ≤ 51, x ≥ 0, y ≥ 0

Graph -3x - 4y < -15 and 5x - 3y ≥ -4

Shade where both hold; the lines cross at (1, 3)

Worksheets · Algebra 2

Algebra 2 systems of inequalities worksheet

Q: Are these systems of inequalities worksheets free?

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.

Q: Do the systems of inequalities worksheets come with an answer key?

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.

Q: Are these systems of inequalities worksheets printable?

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.

Q: What grade are these systems of inequalities worksheets for?

Algebra 2 is usually taken around 10th or 11th 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.

Q: Can I get a different version of the same systems of inequalities worksheet?

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.

Q: Is this a free alternative to Kuta Software for systems of inequalities?

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.

Q: Can I choose how hard the systems of inequalities questions are?

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.

Q: What kinds of question are on a systems of inequalities worksheet?

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 fraction slopes, standard form, vertical and horizontal boundaries. Tick as many as you want and set how many of each, or let it spread them evenly.

Q: What do students get wrong with systems of inequalities?

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.

Shade each inequality, and the solution is the part shaded twice. Everything from the single-inequality sheet applies, done twice and then intersected.

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



Sigma Prep · sigmaprep.io/worksheets Name Systems of Inequalities Date Period Graph each system, or write it as asked. Graph $y \leq 3 x - 12$ and $y \geq x - 8$ Write the system of inequalities shown by the graph. You need at least 13 items: x pens at $$ 1$ each and y notebooks at $$ 4$ each, spending no more than $$ 46$ . Write the system. Graph $y \leq 3 x + 17$ and $y < - x - 7$ Write the system of inequalities shown by the graph. You need at least 8 items: x cupcakes at $$ 2$ each and y cookies at $$ 4$ each, spending no more than $$ 21$ . Write the system. Graph $y \geq - x - 4$ and $y < 3 x - 8$ Write the system of inequalities shown by the graph. You need at least 18 items: x pens at $$ 5$ each and y notebooks at $$ 2$ each, spending no more than $$ 96$ . Write the system. Graph $y < 3 x - 15$ and $y < - 3 x + 15$ Write the system of inequalities shown by the graph. You need at least 13 items: x pens at $$ 1$ each and y notebooks at $$ 7$ each, spending no more than $$ 83$ . Write the system. Graph $y > - x + 2$ and $y > 2 x + 11$ Write the system of inequalities shown by the graph. You need at least 9 items: x cupcakes at $$ 4$ each and y cookies at $$ 3$ each, spending no more than $$ 38$ . Write the system. Graph $y \geq - 3 x - 22$ and $y \leq x + 2$ Write the system of inequalities shown by the graph. You need at least 19 items: x cupcakes at $$ 2$ each and y cookies at $$ 8$ each, spending no more than $$ 143$ . Write the system. Graph $y < - x - 5$ and $y > - 3 x - 11$ Write the system of inequalities shown by the graph. 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.

Graph $y \leq 3 x - 12$ and $y \geq x - 8$
Write the system of inequalities shown by the graph.
You need at least 13 items: x pens at $$ 1$ each and y notebooks at $$ 4$ each, spending no more than $$ 46$ . Write the system.
Graph $y \leq 3 x + 17$ and $y < - x - 7$
Write the system of inequalities shown by the graph.
You need at least 8 items: x cupcakes at $$ 2$ each and y cookies at $$ 4$ each, spending no more than $$ 21$ . Write the system.
Graph $y \geq - x - 4$ and $y < 3 x - 8$
Write the system of inequalities shown by the graph.
You need at least 18 items: x pens at $$ 5$ each and y notebooks at $$ 2$ each, spending no more than $$ 96$ . Write the system.
Graph $y < 3 x - 15$ and $y < - 3 x + 15$
Write the system of inequalities shown by the graph.
You need at least 13 items: x pens at $$ 1$ each and y notebooks at $$ 7$ each, spending no more than $$ 83$ . Write the system.
Graph $y > - x + 2$ and $y > 2 x + 11$
Write the system of inequalities shown by the graph.
You need at least 9 items: x cupcakes at $$ 4$ each and y cookies at $$ 3$ each, spending no more than $$ 38$ . Write the system.
Graph $y \geq - 3 x - 22$ and $y \leq x + 2$
Write the system of inequalities shown by the graph.
You need at least 19 items: x cupcakes at $$ 2$ each and y cookies at $$ 8$ each, spending no more than $$ 143$ . Write the system.
Graph $y < - x - 5$ and $y > - 3 x - 11$
Write the system of inequalities shown by the graph.



Sigma Prep · sigmaprep.io/worksheets Name Systems of InequalitiesAnswer keyVersion 1 Date Period Graph each system, or write it as asked. Graph $y \leq 3 x - 12$ and $y \geq x - 8$ the overlap, cornered at $(2, - 6)$ Write the system of inequalities shown by the graph. $y > - 2 x - 3$ , $y > - x - 1$ You need at least 13 items: x pens at $$ 1$ each and y notebooks at $$ 4$ each, spending no more than $$ 46$ . Write the system. $x + y \geq 13$ , $x + 4 y \leq 46$ , $x \geq 0$ , $y \geq 0$ Graph $y \leq 3 x + 17$ and $y < - x - 7$ the overlap, cornered at $(- 6, - 1)$ Write the system of inequalities shown by the graph. $y > - \frac{1}{2} x + 2$ , $y \leq - x + 4$ You need at least 8 items: x cupcakes at $$ 2$ each and y cookies at $$ 4$ each, spending no more than $$ 21$ . Write the system. $x + y \geq 8$ , $2 x + 4 y \leq 21$ , $x \geq 0$ , $y \geq 0$ Graph $y \geq - x - 4$ and $y < 3 x - 8$ the overlap, cornered at $(1, - 5)$ Write the system of inequalities shown by the graph. $y \geq - x + 4$ , $y \leq \frac{1}{2} x + 4$ You need at least 18 items: x pens at $$ 5$ each and y notebooks at $$ 2$ each, spending no more than $$ 96$ . Write the system. $x + y \geq 18$ , $5 x + 2 y \leq 96$ , $x \geq 0$ , $y \geq 0$ Graph $y < 3 x - 15$ and $y < - 3 x + 15$ the overlap, cornered at $(5, 0)$ Write the system of inequalities shown by the graph. $y \geq - x - 1$ , $y \geq x - 2$ You need at least 13 items: x pens at $$ 1$ each and y notebooks at $$ 7$ each, spending no more than $$ 83$ . Write the system. $x + y \geq 13$ , $x + 7 y \leq 83$ , $x \geq 0$ , $y \geq 0$ Graph $y > - x + 2$ and $y > 2 x + 11$ the overlap, cornered at $(- 3, 5)$ Write the system of inequalities shown by the graph. $y \geq - x + 1$ , $y \geq \frac{1}{2} x - 3$ You need at least 9 items: x cupcakes at $$ 4$ each and y cookies at $$ 3$ each, spending no more than $$ 38$ . Write the system. $x + y \geq 9$ , $4 x + 3 y \leq 38$ , $x \geq 0$ , $y \geq 0$ Graph $y \geq - 3 x - 22$ and $y \leq x + 2$ the overlap, cornered at $(- 6, - 4)$ Write the system of inequalities shown by the graph. $y \geq 2 x - 3$ , $y < x - 1$ You need at least 19 items: x cupcakes at $$ 2$ each and y cookies at $$ 8$ each, spending no more than $$ 143$ . Write the system. $x + y \geq 19$ , $2 x + 8 y \leq 143$ , $x \geq 0$ , $y \geq 0$ Graph $y < - x - 5$ and $y > - 3 x - 11$ the overlap, cornered at $(- 3, - 2)$ Write the system of inequalities shown by the graph. $y > x - 1$ , $y \leq \frac{1}{2} x - 2$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Systems of InequalitiesAnswer keyVersion 1

Date Period

Graph each system, or write it as asked.

Graph $y \leq 3 x - 12$ and $y \geq x - 8$ the overlap, cornered at $(2, - 6)$
Write the system of inequalities shown by the graph. $y > - 2 x - 3$ , $y > - x - 1$
You need at least 13 items: x pens at $$ 1$ each and y notebooks at $$ 4$ each, spending no more than $$ 46$ . Write the system. $x + y \geq 13$ , $x + 4 y \leq 46$ , $x \geq 0$ , $y \geq 0$
Graph $y \leq 3 x + 17$ and $y < - x - 7$ the overlap, cornered at $(- 6, - 1)$
Write the system of inequalities shown by the graph. $y > - \frac{1}{2} x + 2$ , $y \leq - x + 4$
You need at least 8 items: x cupcakes at $$ 2$ each and y cookies at $$ 4$ each, spending no more than $$ 21$ . Write the system. $x + y \geq 8$ , $2 x + 4 y \leq 21$ , $x \geq 0$ , $y \geq 0$
Graph $y \geq - x - 4$ and $y < 3 x - 8$ the overlap, cornered at $(1, - 5)$
Write the system of inequalities shown by the graph. $y \geq - x + 4$ , $y \leq \frac{1}{2} x + 4$
You need at least 18 items: x pens at $$ 5$ each and y notebooks at $$ 2$ each, spending no more than $$ 96$ . Write the system. $x + y \geq 18$ , $5 x + 2 y \leq 96$ , $x \geq 0$ , $y \geq 0$
Graph $y < 3 x - 15$ and $y < - 3 x + 15$ the overlap, cornered at $(5, 0)$
Write the system of inequalities shown by the graph. $y \geq - x - 1$ , $y \geq x - 2$
You need at least 13 items: x pens at $$ 1$ each and y notebooks at $$ 7$ each, spending no more than $$ 83$ . Write the system. $x + y \geq 13$ , $x + 7 y \leq 83$ , $x \geq 0$ , $y \geq 0$
Graph $y > - x + 2$ and $y > 2 x + 11$ the overlap, cornered at $(- 3, 5)$
Write the system of inequalities shown by the graph. $y \geq - x + 1$ , $y \geq \frac{1}{2} x - 3$
You need at least 9 items: x cupcakes at $$ 4$ each and y cookies at $$ 3$ each, spending no more than $$ 38$ . Write the system. $x + y \geq 9$ , $4 x + 3 y \leq 38$ , $x \geq 0$ , $y \geq 0$
Graph $y \geq - 3 x - 22$ and $y \leq x + 2$ the overlap, cornered at $(- 6, - 4)$
Write the system of inequalities shown by the graph. $y \geq 2 x - 3$ , $y < x - 1$
You need at least 19 items: x cupcakes at $$ 2$ each and y cookies at $$ 8$ each, spending no more than $$ 143$ . Write the system. $x + y \geq 19$ , $2 x + 8 y \leq 143$ , $x \geq 0$ , $y \geq 0$
Graph $y < - x - 5$ and $y > - 3 x - 11$ the overlap, cornered at $(- 3, - 2)$
Write the system of inequalities shown by the graph. $y > x - 1$ , $y \leq \frac{1}{2} x - 2$

How to do these

Graph $y \geq x - 1$ and $y < - 2 x + 4$

Shade above the first line, solid.
Shade below the second, dashed.
The solution is where both shadings land.

The answer is the overlap of the two regions.

Where students go wrong

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.

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 $y \geq x + 1$ and $y \geq - 2 x + 1$ → the overlap, cornered at $(0, 1)$
Write the system from a graph: Write the system of inequalities shown by the graph. → $y > - \frac{1}{2} x$ , $y \leq x + 1$
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 $$ 51$ . Write the system. → $x + y \geq 14$ , $3 x + 7 y \leq 51$ , $x \geq 0$ , $y \geq 0$
Name a point: level lines, standard form, strips: Name one point in the solution:
$y \geq - 1$
$y \leq 2 x + 6$ → e.g. $(1, 5)$
Fraction slopes, standard form, vertical and horizontal boundaries: Graph $- 3 x - 4 y < - 15$ and $5 x - 3 y \geq - 4$ → Shade where both hold; the lines cross at $(1, 3)$

Questions about these worksheets