Linear Inequalities Geeksforgeeks

Bonisiwe Shabane

-Oct 31, 2025, 8:13 AM

Linear Inequalities are mathematical statements that show the relationship between two expressions using inequality symbols instead of an equal sign. They are similar to a linear equation, but instead of “=”, they use symbols like: Linear inequalities are formed by combining linear algebraic expressions with inequalities. A linear algebraic expression has a degree of one, meaning it involves terms of degree one. In linear inequalities, at least one quantity being compared must be a polynomial. Following are some examples of linear inequalities with their meaning:

In mathematics, inequality occurs when a non-equal comparison is made between two mathematical expressions or two numbers. In general, inequalities can be either numerical inequality or algebraic inequality or a combination of both. Linear inequalities are inequalities that involve at least one linear algebraic expression, that is, a polynomial of degree 1 is compared with another algebraic expression of degree less than or equal to 1. There are several ways to represent various kinds of linear inequalities. In this article, let us learn about linear inequalities, solving linear inequalities, graphing linear inequalities. Linear inequalities are defined as expressions in which two linear expressions are compared using the inequality symbols.

The five symbols that are used to represent the linear inequalities are listed below: We need to note that if, p < q, then p is some number that is strictly less than q. If p ≤ q, then it means that p is some number that is either strictly less than q or is exactly equal to q. Likewise, the same applies to the remaining two inequalities > (greater than) and ≥ (greater than or equal to). Now, Let's say we have a linear inequality, 3x - 4 < 20. In this case LHS < RHS.

We can see that the expression on the left-hand side, that is, 3x - 4 is in fact lesser than the number on the right-hand side, which is 20. We can represent this inequality pictorially on a weighing scale as: \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \) \( \newcommand{\dsum}{\displaystyle\sum\limits} \) \( \newcommand{\dint}{\displaystyle\int\limits} \)

\( \newcommand{\dlim}{\displaystyle\lim\limits} \) Linear inequalities are fundamental concepts in algebra that help us understand and solve a wide range of real-world problems. These inequalities express the relationship between two expressions using inequality symbols like <, ≤, >, and ≥. Learning to solve linear inequalities involves understanding how to manipulate these expressions to isolate the variable and determine the range of possible solutions. In this article, we will explore various practice questions on linear inequalities to enhance your understanding and problem-solving skills. We will cover single-variable inequalities, systems of linear inequalities, and graphical representations of these solutions.

Each section includes step-by-step explanations to help you grasp the methods used in solving different types of inequalities. Linear inequalities are mathematical expressions involving linear functions that use inequality symbols (such as <, ≤, >, or ≥) instead of an equality sign. These inequalities describe a relationship where one side of the expression is not strictly equal to the other but is either less than, greater than, or equal to (depending on the inequality used). Problem 1: Solve the inequality: x - 5 ≤ 3. Problem 2: Solve the inequality: 4x - 7 > 9 Share this post with a friend and contribute to making knowledge accessible to everyone.

An inequality is a mathematical statement involving algebraic expressions for which we seek the values of the variables that make the inequality true. In general, an inequality between two algebraic expressions \( A(x) \) and \( B(x) \) is defined as the relation: Solving an inequality involves determining the solution set, that is, all the values of \( x \) that satisfy the previous inequality. The solutions of inequalities are subsets of \( \mathbb{R} \) defined as intervals. Given two real numbers \( a \) and \( b \) with \( a < b \), a bounded interval is defined as the set of real numbers between \( a \) and \(... An unbounded interval is the set of numbers that either precede \( a \) or follow \( a \).

If the endpoints of a bounded interval are included, the interval is called closed, otherwise, it is called open. The degree of an inequality corresponds to the degree of the polynomial \( P(x) \) obtained by rewriting the inequality in the form \( P(x) > 0 \). A linear or first-degree inequality is defined in the general form: Linear inequalities are math statements where two numbers or expressions are compared using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Linear inequalities (also called linear inequations) show a comparison between two algebraic expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal... These involve just one variable, and the expressions are compared using inequality signs.

Example: 3x + 5 < 10 — This means that the value of 3x + 5 must be less than 10. These include two variables, usually x and y, and use inequality signs to relate both. Example: 3x + 5y ≥ 20 — This means that the total of 3x + 5y must be at least 20. The following symbols are used to represent linear inequalities :

Linear Inequalities Geeksforgeeks

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\( \newcommand{\dlim}{\displaystyle\lim\limits} \) Linear Inequalities Are Fundamental Concepts In Algebra