Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -2x + 8 ≤ 16

Verified step by step guidance

1. Start by isolating the variable term on one side of the inequality. Subtract 8 from both sides to get: \(-2x + 8 - 8 \leq 16 - 8\) which simplifies to \(-2x \leq 8\).
2. Next, solve for \(x\) by dividing both sides of the inequality by \(-2\). Remember, when dividing or multiplying an inequality by a negative number, the inequality sign must be reversed. So, dividing both sides by \(-2\) gives: \(x \geq \frac{8}{-2}\).
3. Simplify the right side of the inequality: \(x \geq -4\).
4. Interpret the inequality \(x \geq -4\) to understand the solution set. This means \(x\) includes all real numbers greater than or equal to \(-4\).
5. Express the solution set in interval notation. Since \(x\) is greater than or equal to \(-4\), the interval notation is \([-4, \infty)\).

Key Concepts

Solving Linear Inequalities

A linear inequality involves an inequality sign (<, ≤, >, ≥) with a linear expression. To solve it, isolate the variable on one side by performing inverse operations, similar to solving equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.

Interval Notation

Interval notation is a way to represent the solution set of inequalities using intervals. It uses parentheses () for strict inequalities and brackets [] for inclusive inequalities, indicating the range of values that satisfy the inequality.

Properties of Inequalities

Understanding how inequalities behave under addition, subtraction, multiplication, and division is crucial. Notably, multiplying or dividing both sides by a negative number reverses the inequality sign, which is essential to correctly solve and express the solution.