The inequality uses a strict sign, , so points on the line are not solutions.

System of Inequalities

Sommaire

In short: A system of inequalities pairs several inequality conditions on shared variables, and every point of its solution region obeys all the conditions together.

Algebra 1AlgebraA-REI.D.12

What is System of Inequalities?

A system of inequalities expresses several rules at once, like a school store budget of no more than $24 combined with a minimum of 4 items, and each rule keeps one side of a boundary line.

Draw all the boundary lines in one coordinate plane. A solid boundary means points on the line count ($\le$ or $\ge$); a dashed boundary means they do not (strict signs). The overlap of all the shaded half-planes is the solution region, and every point inside it makes every inequality true.

How to spot it

A story has two or more limits, such as a budget cap and a minimum quantity.
The problem lists inequalities with x and y and asks for a graph or for the solution region.
You must decide whether a specific ordered pair works in every inequality.
The graph shows two boundary lines and a shaded area where the shadings overlap.

The method

Rewrite each inequality with y alone (flip the sign if you divide by a negative) so the boundary is easy to graph.
Draw each boundary line, solid for $\le$ and $\ge$, dashed for strict signs.
Shade the correct side of each line: above for $y >$ or $y \ge$, below for $y <$ or $y \le$.
Mark the overlap as the solution, then test one point inside it in every original inequality.

The green overlap satisfies both inequalities; the lines cross at (3, 3).

System of Inequalities: solved example

Problem

At a school store, pens cost $2 and notebooks cost $3. Hana can spend at most $24 and wants at least 4 items in all. With x pens and y notebooks, write the system and decide whether buying 3 pens and 4 notebooks works.

Solution

Spending: $2x + 3y \le 24$. Item count: $x + y \ge 4$. Also $x \ge 0$ and $y \ge 0$.

Test $(3, 4)$: $2(3) + 3(4) = 18 \le 24$ is true, and $3 + 4 = 7 \ge 4$ is true. Both hold, so yes, 3 pens and 4 notebooks work; the point lies in the solution region.

Try it

Question 1

Is $(2, 1)$ a solution of $y > x – 3$ and $y \le 4$?

Show the answer

First: $1 > -1$ is true. Second: $1 \le 4$ is true. Yes.

Question 2

Why would a boundary line be dashed?

Show the answer

The inequality uses a strict sign, $<$ or $>$, so points on the line are not solutions.

System of Inequalities vs. System of Equations

A system of equations usually pins down one point where the lines cross. A system of inequalities has a whole region of solutions, often infinitely many points, and the crossing point may or may not even belong to it. Think area versus single location.

System of Equations

Keep going

Lessons

Linear Inequalities in Two Variables and Systems

Worksheets and quizzes

Related words

Whole dictionary: A–Z · Algebra 1 · Algebra

Voter.. page