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 constants inside the parentheses on the left side: multiply 5 by (x - 2) and -3 by (x + 4), resulting in \$5x - 10\( and \)-3x - 12$ respectively.
2. Combine like terms on the left side by adding \$5x\( and \)-3x\(, and adding \)-10\( and \)-12$, to simplify the left side expression.
3. Rewrite the inequality with the simplified left side and the right side as is: \(\text{(simplified left side)} \geq 2x - 20\).
4. Get all variable terms on one side and constants on the other by subtracting \$2x$ from both sides and adding or subtracting constants accordingly.
5. Solve for \(x\) by isolating it on one side, then express the solution set in interval notation and prepare to graph it on a number line.

Key Concepts

Solving Linear Inequalities

Linear inequalities involve expressions with variables raised to the first power and inequality symbols (>, <, ≥, ≤). Solving them requires isolating the variable on one side by performing algebraic operations, similar to solving linear equations, but with attention to inequality rules.

Properties of Inequalities

When solving inequalities, adding or subtracting the same number on both sides preserves the inequality direction. However, multiplying or dividing both sides by a negative number reverses the inequality sign. Understanding these properties is essential to correctly solve and interpret inequalities.

Interval Notation and Graphing Solutions

Interval notation expresses solution sets as intervals on the number line, using parentheses for strict inequalities and brackets for inclusive inequalities. Graphing these solutions visually represents the range of values satisfying the inequality, aiding in comprehension and communication of the solution.