Solution of a Linear Inequality
A solution of a linear inequality is any ordered pair (x, y) that makes the inequality true when substituted for x and y. Unlike linear equations — which have a line of solutions — linear inequalities have a half-plane of solutions: the entire region on one side of a boundary line. Chapter 4 of OpenStax Elementary Algebra 2E introduces this concept as a direct extension of solving linear equations. Verifying a point is a solution simply requires substituting its coordinates and checking whether the inequality holds, which is the same process used to check a solution to an equation.
Key Concepts
Property An ordered pair $(x, y)$ is a solution of a linear inequality if the inequality is true when we substitute the values of $x$ and $y$.
Examples To check if $(1, 4)$ is a solution to $y 3x 1$, substitute: $4 3(1) 1$ becomes $4 2$. This is true, so $(1, 4)$ is a solution. To check if $(2, 1)$ is a solution to $2x + y \leq 3$, substitute: $2(2) + ( 1) \leq 3$ becomes $3 \leq 3$. This is true, so $(2, 1)$ is a solution. To check if $(0, 0)$ is a solution to $x 5y 1$, substitute: $0 5(0) 1$ becomes $0 1$. This is false, so $(0, 0)$ is not a solution.
Explanation A solution is any point $(x, y)$ that makes the inequality true. Unlike equations that have solutions on a line, inequalities have solutions in a whole shaded region. Any point in that region works!
Common Questions
What is a solution of a linear inequality?
A solution is any ordered pair (x, y) that satisfies the inequality when substituted. For example, (1, 2) is a solution of y > x because 2 > 1 is true.
How do I check if a point is a solution to a linear inequality?
Substitute the x and y coordinates into the inequality and evaluate. If the result is a true statement, the point is a solution. If it is false, it is not.
What is the difference between solutions of an equation and an inequality?
A linear equation in two variables has infinitely many solutions that form a line. A linear inequality has infinitely many solutions that form a half-plane — all points on one side of the boundary line.
Does the boundary line count as part of the solution?
It depends on the inequality symbol. If the symbol is ≤ or ≥, the boundary line is included (shown as a solid line). If the symbol is < or >, the boundary is excluded (shown as a dashed line).
When do students learn about solutions of linear inequalities?
This is an algebra 1 topic taught alongside graphing linear equations, covered in OpenStax Elementary Algebra 2E Chapter 4: Graphs.
What is a common mistake when identifying solutions of an inequality?
Testing a point that lies exactly on the boundary line — this only works for ≤ or ≥, not for strict inequalities. Always test a point clearly inside the potential solution region.
How does this concept connect to graphing inequalities?
When you graph a linear inequality, shading the solution region represents all ordered pairs that satisfy the inequality. The shade is on the side of the boundary line where a test point is true.