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: \(|2x + \frac{1}{3}| + 1 < 4\) becomes \(|2x + \frac{1}{3}| < 3\).
2. Recall that for an inequality of the form \(|A| < B\) where \(B > 0\), the solution is \(-B < A < B\). Apply this to get: \(-3 < 2x + \frac{1}{3} < 3\).
3. Next, solve the compound inequality by isolating \(x\). Start with the left part: \(-3 < 2x + \frac{1}{3}\). Subtract \(\frac{1}{3}\) from both sides to get \(-3 - \frac{1}{3} < 2x\).
4. Then, divide both sides of the inequality by 2 to solve for \(x\): \(\frac{-3 - \frac{1}{3}}{2} < x\).
5. Repeat the process for the right part of the inequality: \$2x + \frac{1}{3} < 3\(. Subtract \)\frac{1}{3}\( from both sides and then divide by 2 to find the upper bound for \)x$. Combine both results to express the solution as an interval.

Key Concepts

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, you consider the definition of absolute value as distance from zero, leading to two separate inequalities that capture the positive and negative scenarios.

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 subtracting or adding constants and dividing by coefficients, ensuring the inequality is in a standard form for applying the absolute value rules.

Solving Linear Inequalities

Once the absolute value is removed, solving the resulting linear inequalities requires applying standard algebraic techniques such as adding, subtracting, multiplying, or dividing both sides by constants, while carefully considering the direction of the inequality when multiplying or dividing by negative numbers.