Solve each inequality. Give the solution set in interval notation. 5| x + 1 | > 10

Verified step by step guidance

1. Start by isolating the absolute value expression. Divide both sides of the inequality by 5 to get \(|x + 1| > 2\).
2. Recall that an inequality of the form \(|A| > B\) implies two separate inequalities: \(A > B\) or \(A < -B\).
3. Apply this to the expression \(|x + 1| > 2\), resulting in two inequalities: \(x + 1 > 2\) and \(x + 1 < -2\).
4. Solve each inequality separately. For \(x + 1 > 2\), subtract 1 from both sides to get \(x > 1\). For \(x + 1 < -2\), subtract 1 from both sides to get \(x < -3\).
5. Combine the solutions from both inequalities. The solution set in interval notation is \((-\infty, -3) \cup (1, \infty)\).

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions that contain absolute values, which measure the distance of a number from zero on the number line. To solve an inequality like |x + 1| > a, where a is a positive number, we split it into two separate inequalities: x + 1 > a and x + 1 < -a. This allows us to find the range of values for x that satisfy the inequality.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the 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.

Solving Inequalities

Solving inequalities involves finding the values of a variable that make the inequality true. This process often includes isolating the variable on one side of the inequality sign and may require reversing the inequality sign when multiplying or dividing by a negative number. The solution is typically expressed in interval notation to clearly indicate the set of valid solutions.