In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 8x - 11 ≤ 3x - 13

Verified step by step guidance

1. Start by isolating the variable term on one side of the inequality. Subtract $3x$ from both sides: $8x-11-3x\le \; -13$.
2. Combine like terms on the left-hand side: $5x-11\le \; -13$.
3. Next, isolate the term with the variable by adding $11$ to both sides: $5x\le \; -13+11$.
4. Simplify the right-hand side: $5x\le \; -2$.
5. Finally, divide both sides of the inequality by $5$ to solve for $x$: $x\le \; -2/5$. Express the solution in interval notation and graph it on a number line.

Key Concepts

Linear Inequalities

Linear inequalities are mathematical expressions that involve a linear function and an inequality sign (such as ≤, ≥, <, or >). They represent a range of values rather than a single solution. Understanding how to manipulate these inequalities is crucial for solving them, as it involves similar techniques to solving linear equations, but with special attention to the direction of the inequality when multiplying or dividing by negative numbers.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5. This notation is essential for expressing the solution sets of inequalities succinctly.

Graphing on a Number Line

Graphing solution sets on a number line visually represents the range of values that satisfy an inequality. This involves marking points and using open or closed circles to indicate whether endpoints are included or excluded. Understanding how to accurately depict these solutions helps in interpreting the results of inequalities and provides a clear visual reference for the solution set.