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. 3x - 7 ≥ 13

Verified step by step guidance

1. Start by isolating the variable term on one side of the inequality. Add 7 to both sides to get: \(3x - 7 + 7 \geq 13 + 7\).
2. Simplify the inequality: \(3x \geq 20\).
3. Next, solve for \(x\) by dividing both sides by 3: \(x \geq \frac{20}{3}\).
4. Express the solution in interval notation. Since \(x\) is greater than or equal to \(\frac{20}{3}\), the interval is: \([\frac{20}{3}, \infty)\).
5. Graph the solution on a number line. Draw a closed circle at \(\frac{20}{3}\) and shade the line to the right, indicating all numbers greater than or equal to \(\frac{20}{3}\).

Key Concepts

Linear Inequalities

A linear inequality is a mathematical statement that compares a linear expression to a value using inequality symbols such as <, >, ≤, or ≥. Unlike equations, inequalities express a range of possible solutions rather than a single value. Understanding how to manipulate these inequalities is crucial for finding solution sets.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the number line. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval [a, b) includes 'a' but excludes 'b', which is essential for expressing solution sets of inequalities.

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. This visual representation helps in understanding the solution set and its implications in real-world contexts.