Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9.5x +2 ≤ -48

Verified step by step guidance

1. Start by isolating the variable term on one side of the inequality. Subtract 2 from both sides to get: \(5x \leq -48 - 2\).
2. Simplify the right side of the inequality: \(5x \leq -50\).
3. To solve for \(x\), divide both sides of the inequality by 5: \(x \leq \frac{-50}{5}\).
4. Simplify the division on the right side: \(x \leq -10\).
5. Express the solution set in interval notation: \((-\infty, -10]\).

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 often requires similar techniques as solving 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 include adding, subtracting, multiplying, or dividing both sides by a constant. It is crucial to remember 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.