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. 5(x - 2) - 3(x + 4) ≥ 2x - 20

Verified step by step guidance

1. Distribute the 5 into the terms inside the parentheses: 5(x - 2) becomes 5x - 10.
2. Distribute the -3 into the terms inside the parentheses: -3(x + 4) becomes -3x - 12.
3. Combine like terms on the left side of the inequality: 5x - 3x - 10 - 12.
4. Simplify the left side: 2x - 22.
5. Rearrange the inequality to isolate x: 2x - 22 ≥ 2x - 20.

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. Solving a linear inequality involves isolating the variable on one side of the inequality, similar to solving an equation, but requires 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 but not 2. This notation is essential for expressing solution sets of inequalities succinctly.

Graphing on a Number Line

Graphing on a number line involves visually representing the solution set of an inequality. Each solution is marked on the line, with open circles indicating excluded endpoints and closed circles indicating included endpoints. This graphical representation helps to quickly convey the range of solutions and is a useful tool for understanding the behavior of inequalities in a visual context.