What is Linear Inequality Solution?

A linear inequality solution is the set of all values that, when substituted for the variable(s) in a linear inequality, make the inequality a true statement.

Understanding Linear Inequalities

Linear inequalities are similar to linear equations but use inequality symbols (>, <, ≥, ≤) instead of an equals sign (=). Solving them involves finding the range of values that satisfy the inequality. The steps are similar to solving linear equations, but with a crucial difference when multiplying or dividing by a negative number.

Steps to Find the Solution

The process of solving a single linear inequality closely mirrors that of solving linear equations. Here's a breakdown of the key steps:

Simplify Both Sides: Combine like terms and use the distributive property to eliminate parentheses on each side of the inequality.
Isolate the Variable Term: Manipulate the inequality to get all terms containing the variable on one side and all constant terms on the other. This is done using addition or subtraction.
Solve for the Variable: Multiply or divide both sides of the inequality by the coefficient of the variable to isolate the variable. Important: If you multiply or divide by a negative number, you must reverse the direction of the inequality sign.

Representing the Solution

The solution to a linear inequality can be represented in several ways:

Inequality Notation: This is the most common way, expressing the solution as a range of values. For example, x > 3 means all values of x greater than 3.
Number Line: The solution can be graphed on a number line. An open circle indicates that the endpoint is not included in the solution (for > and <), while a closed circle indicates that the endpoint is included (for ≥ and ≤). The line extends in the direction of all values that satisfy the inequality.
Interval Notation: This notation uses parentheses and brackets to represent the solution. Parentheses () indicate that the endpoint is not included, while brackets [] indicate that the endpoint is included. Infinity (∞) and negative infinity (-∞) are always enclosed in parentheses. For example, (3, ∞) represents all numbers greater than 3.

Examples

Example 1:

Solve the inequality: 2x + 3 < 7

Subtract 3 from both sides: 2x < 4
Divide both sides by 2: x < 2

Solution: x < 2 (Inequality Notation)
Number Line: A number line with an open circle at 2 and the line shaded to the left.
Interval Notation: (-∞, 2)

Example 2:

Solve the inequality: -3x + 1 ≥ 10

Subtract 1 from both sides: -3x ≥ 9
Divide both sides by -3 (and reverse the inequality sign!): x ≤ -3

Solution: x ≤ -3 (Inequality Notation)
Number Line: A number line with a closed circle at -3 and the line shaded to the left.
Interval Notation: (-∞, -3]

In summary, the solution to a linear inequality is the set of all values for the variable that make the inequality true. Finding the solution involves isolating the variable using algebraic operations, keeping in mind the rule about reversing the inequality sign when multiplying or dividing by a negative number. The solution can be expressed in inequality notation, graphically on a number line, or in interval notation.