Solving Linear Inequalities

Understanding Linear Inequalities

Linear inequalities are mathematical statements that relate linear expressions using inequality symbols such as <, >, ≤, or ≥. Solving these inequalities involves finding all values of the variable that make the inequality true.

• Linear inequality: An inequality that involves a linear expression, such as .

• Solution set: The set of all real numbers that satisfy the inequality.

• Interval notation: A concise way to describe sets of numbers on the real number line.

Example Problem: Solving and Graphing a Linear Inequality

Consider the inequality:

• Step 1: Distribute and simplify both sides.

• Step 2: Collect like terms.

Add to both sides:

Subtract $28$ from both sides:

• Step 3: Solve for .

Divide both sides by $2$:

Or, equivalently:

• Step 4: Express the solution set in interval notation.

The solution set is all real numbers less than or equal to $2$:

Interval notation:

Graphing the Solution Set on a Number Line

To graph on a number line:

• Draw a solid dot or bracket at $2 is included (since the inequality is "less than or equal to").

• Shade the number line to the left of $2.

Table: Solution Set Representation

Key Points

• Always reverse the inequality sign when multiplying or dividing both sides by a negative number.

• Interval notation uses parentheses for open intervals (not including the endpoint) and brackets for closed intervals (including the endpoint).

• No solution is denoted by the empty set symbol or .

Example

Problem: Solve

• Add $5

• Divide by $3x > 4$

• Interval notation:

Additional info: The process of solving linear inequalities is foundational for understanding more advanced topics in algebra and precalculus, such as systems of inequalities and absolute value inequalities.