Algebra 1 Linear Inequalities in Two Variables and Systems Worksheet (PDF)

Algebra 1 math worksheet: Linear Inequalities in Two Variables and Systems – Maths PDF

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This linear inequalities two variables worksheet for Algebra 1 covers linear inequalities in two variables and systems (Common Core A-REI.D.12, A-CED.A.3), with a free PDF to print.

Linear inequalities two variables worksheet (Algebra 1) – first page of the printable PDF

This worksheet has 20 problems on linear inequalities in two variables and systems, aligned to Common Core A-REI.D.12 and A-CED.A.3. You will test points, graph half-planes with solid or dashed boundaries, solve systems of inequalities, and decide which combinations fit real constraints. Early problems check one skill at a time; the final six put the skill inside real situations. Write each step on paper before you look at the answer key.

Algebra 1 · Common Core State Standards: A-REI.D.12, A-CED.A.3

Linear inequalities two variables worksheet: exercises 1 to 10

Exercise 1 – Test a point

Is $(2, 1)$ a solution of $y < x + 2$?

Exercise 2 – Not a solution

Is $(0, -2)$ a solution of $y \ge 2x + 3$?

Exercise 3 – Solid or dashed

State whether the boundary line is solid or dashed: (a) $y \ge 2x – 1$ (b) $y < -x + 4$ (c) $y > 3$.

Exercise 4 – Graph y > x + 1

Graph $y > x + 1$.

Exercise 5 – Graph with a solid line

Sketch the graph of $4x + y \ge 4$ and state which side of the boundary is shaded.

Exercise 6 – Vertical boundary

Graph $x \ge 2$.

Exercise 7 – Horizontal boundary

Graph $y < 3$.

Exercise 8 – Write from a graph

A shaded half-plane.

Write the inequality graphed.

Exercise 9 – Standard form inequality

Graph $2x + 3y \le 12$.

Exercise 10 – Negative coefficient

Graph $3x – y > 3$.

Exercises 11 and up

Exercise 11 – Which points work?

Which of the points $(0, 0)$, $(3, 1)$, $(-2, 5)$ are solutions of $x + 2y \le 4$?

Exercise 12 – System of two inequalities

Graph the system $y \ge \frac{1}{2}x – 2$ and $y < -x + 4$.

Exercise 13 – Triangular region

Graph the system $x \ge 0$, $y \ge 0$, and $x + y \le 5$. Is $(2, 2)$ a solution?

Exercise 14 – Check a system

Is $(3, 2)$ a solution of $y > x – 2$ and $y \le 2x$?

Exercise 15 – Write a system from a graph

Shaded region.

Write the system of inequalities shown in the graph.

Exercise 16 – Lunch budget

Prices.

Priya has at most $30 to spend. Wraps cost $5 and juices cost $3. Write an inequality for the numbers of wraps $x$ and juices $y$ and decide whether she can buy 3 wraps and 5 juices.

Exercise 17 – Two jobs

Mei works at most 12 hours a week. She earns $10 per hour tutoring ($x$ hours) and $8 per hour at a store ($y$ hours). She wants at least $90. Write a system and decide whether 6 hours tutoring and 5 hours at the store works. What about 4 and 4?

Exercise 18 – Whole-number combinations

Possible combinations.

A craft kit uses 2 beads of type A and 3 beads of type B per bracelet style, and a student has at most 12 beads in total: $2x + 3y \le 12$ with $x$ and $y$ whole numbers at least 0. How many whole-number pairs $(x, y)$ work?

Exercise 19 – Truck load

A truck can carry at most 280 pounds. Small crates weigh 25 pounds ($x$) and large crates weigh 40 pounds ($y$). Write an inequality. Can it carry 4 small and 4 large crates? 6 small and 4 large?

Exercise 20 – School store

A school store needs to sell at least 10 items and stay under a $24 budget. Pencils cost $1 each ($p$) and notebooks cost $3 each ($n$). Write the system $p + n \ge 10$ and $p + 3n \le 24$. Is $(6, 4)$ a solution? Is $(2, 8)$?

Exercise 21 – Corner of the region

School store region (p across, n up).

Find the corner of the feasible region in the school store problem where the lines $p + n = 10$ and $p + 3n = 24$ meet.

View solution

Key facts behind this linear inequalities two variables worksheet

Key facts

A solution of an inequality in two variables is an ordered pair that makes it true.
The boundary line is solid for ≤ and ≥, and dashed for < and >.
Test a point not on the line to decide which side to shade.
For y > mx + b, shade above; for y < mx + b, shade below.
The solution of a system is the overlap of the shaded regions.

How to use this linear inequalities two variables worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the linear inequalities two variables worksheet as a PDF with the button below and keep it with your notes.

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