Solve each inequality. Give the solution set in interval notation. 5| x + 1 | > 12

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Start by isolating the absolute value expression. Divide both sides of the inequality \(5| x + 1 | > 10\) by 5 to get \(| x + 1 | > 2\).

Recall that the inequality \(| A | > B\) (where \(B > 0\)) means that \(A > B\) or \(A < -B\). Apply this to \(| x + 1 | > 2\) to write two inequalities: \(x + 1 > 2\) or \(x + 1 < -2\).

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 \(x < -3\).

Combine the two solution sets. The solution to the original inequality is all \(x\) such that \(x > 1\) or \(x < -3\).

Express the solution set in interval notation as \((-\infty, -3) \cup (1, \infty)\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable or expression is compared to a number. To solve them, consider the definition of absolute value as distance from zero, leading to two cases: one where the expression inside is greater than the positive number, and one where it is less than the negative of that number.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This often involves dividing or multiplying both sides by a positive number, which does not change the inequality direction, making the inequality easier to analyze and solve.

Interval Notation

Interval notation is a way to represent solution sets of inequalities using intervals on the number line. It uses parentheses for open intervals (excluding endpoints) and brackets for closed intervals (including endpoints), providing a concise and clear way to express all values that satisfy the inequality.