Solve each equation or inequality. | 2x+ 1/3 | +1 < 4

Verified step by step guidance

1. Start by isolating the absolute value expression. Subtract 1 from both sides of the inequality: \(|2x + \frac{1}{3}| < 3\).
2. Recognize that the inequality \(|A| < B\) implies \(-B < A < B\). Apply this to the expression: \(-3 < 2x + \frac{1}{3} < 3\).
3. Solve the left part of the compound inequality: \(-3 < 2x + \frac{1}{3}\). Subtract \(\frac{1}{3}\) from both sides: \(-3 - \frac{1}{3} < 2x\).
4. Solve the right part of the compound inequality: \(2x + \frac{1}{3} < 3\). Subtract \(\frac{1}{3}\) from both sides: \(2x < 3 - \frac{1}{3}\).
5. Divide each part of the compound inequality by 2 to solve for \(x\).

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|. For example, |3| = 3 and |-3| = 3. In equations, absolute values can create two separate cases to consider, as they can be equal to both positive and negative values.

Inequalities

Inequalities express a relationship where one side is not necessarily equal to the other, using symbols like <, >, ≤, or ≥. In this context, the inequality |2x + 1/3| + 1 < 4 indicates that the expression on the left must be less than 4. Solving inequalities often involves isolating the variable and considering the implications of the inequality sign.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. In the context of the given problem, this includes manipulating the equation to isolate the variable x. Techniques such as adding, subtracting, multiplying, or dividing both sides of the equation are commonly used to achieve this goal, especially when dealing with absolute values.