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: \(-2x + 8 - 8 \leq 16 - 8\).
2. Simplify the inequality: \(-2x \leq 8\).
3. To solve for \(x\), divide both sides by \(-2\). Remember that dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign: \(x \geq -4\).
4. Express the solution set in interval notation. Since \(x\) is greater than or equal to \(-4\), the interval is \([-4, \infty)\).
5. Verify the solution by testing a number within the interval, such as \(x = 0\), to ensure it satisfies the original inequality.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They can be represented using symbols such as ≤ (less than or equal to), ≥ (greater than or equal to), < (less than), and > (greater than). Solving inequalities involves finding the values of the variable that make the inequality true, which can include manipulating the expression similarly to equations, but with special attention to the direction of the inequality when multiplying or dividing by negative numbers.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. It uses parentheses and brackets to indicate whether the endpoints are included in the set. For example, (a, b) means all numbers between a and b, excluding a and b, while [a, b] includes both endpoints. This notation is particularly useful for expressing the solution sets of inequalities succinctly.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side of the inequality sign, similar to solving linear equations. This process may require adding, subtracting, multiplying, or dividing both sides by a number. A critical aspect to remember is that if you multiply or divide by a negative number, the direction of the inequality sign must be reversed. The solution is then expressed in interval notation to clearly indicate the range of values that satisfy the inequality.