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 with the given inequality: \(3x - 7 \geq 13\).
2. Add 7 to both sides to isolate the term with \(x\): \(3x - 7 + 7 \geq 13 + 7\), which simplifies to \(3x \geq 20\).
3. Divide both sides of the inequality by 3 to solve for \(x\): \(\frac{3x}{3} \geq \frac{20}{3}\), which simplifies to \(x \geq \frac{20}{3}\).
4. Express the solution set in interval notation: \(\left[ \frac{20}{3}, \infty \right)\), where the square bracket indicates that \(\frac{20}{3}\) is included in the solution.
5. To graph the solution on a number line, draw a solid dot at \(\frac{20}{3}\) and shade the line extending to the right towards positive infinity.

Key Concepts

Solving Linear Inequalities

A linear inequality involves an inequality symbol (>, <, ≥, ≤) with a linear expression. To solve it, isolate the variable on one side by performing inverse operations, similar to solving linear equations, but pay attention to inequality rules, especially when multiplying or dividing by negative numbers.

Interval Notation

Interval notation is a way to represent solution sets of inequalities using parentheses and brackets. Parentheses indicate that an endpoint is not included, while brackets mean it is included. For example, [2, ∞) represents all numbers greater than or equal to 2.

Graphing Solution Sets on a Number Line

Graphing solution sets involves marking the range of values that satisfy the inequality on a number line. Use a solid dot for included endpoints (≥ or ≤) and an open circle for excluded endpoints (> or <). Shade the region representing all solutions to visually display the set.