Solve each inequality. Give the solution set in interval notation. | 3/5 + x | < 1

Verified step by step guidance

1. Identify the inequality involving the absolute value: \(| \frac{3}{5} + x | < 1\).
2. Recall that for an inequality of the form \(|A| < B\), where \(B > 0\), the solution is \(-B < A < B\). Apply this to get: \(-1 < \frac{3}{5} + x < 1\).
3. Solve the compound inequality by isolating \(x\). First, subtract \(\frac{3}{5}\) from all parts: \(-1 - \frac{3}{5} < x < 1 - \frac{3}{5}\).
4. Simplify the expressions on both sides by finding a common denominator and performing the subtraction.
5. Write the solution set for \(x\) in interval notation based on the simplified inequality.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value bars, representing distance from zero. To solve |A| < B, where B > 0, rewrite it as a double inequality: -B < A < B. This approach helps isolate the variable and find the range of values satisfying the inequality.

Solving Linear Inequalities

Solving linear inequalities requires isolating the variable on one side by performing inverse operations like addition, subtraction, multiplication, or division. When multiplying or dividing by a negative number, the inequality sign reverses. The solution is often expressed as an interval or inequality.

Interval Notation

Interval notation is a concise way to represent sets of numbers between two endpoints. Parentheses () indicate that endpoints are not included (open interval), while brackets [] mean endpoints are included (closed interval). It is commonly used to express solution sets of inequalities.