Solve each equation or inequality. | 5x + 1/2 | -2 < 5

Verified step by step guidance

1. Start by isolating the absolute value expression. Add 2 to both sides of the inequality to get: \(| 5x + \frac{1}{2} | < 7\).
2. Recall that for an inequality of the form \(|A| < B\), where \(B > 0\), the solution is \(-B < A < B\). Apply this to get: \(-7 < 5x + \frac{1}{2} < 7\).
3. Next, solve the compound inequality by subtracting \(\frac{1}{2}\) from all parts: \(-7 - \frac{1}{2} < 5x < 7 - \frac{1}{2}\).
4. Simplify the expressions on both sides: \(-\frac{15}{2} < 5x < \frac{13}{2}\).
5. Finally, divide all parts of the inequality by 5 to isolate \(x\): \(-\frac{15}{10} < x < \frac{13}{10}\). Simplify the fractions if possible.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable 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 to solve.

Isolating the Absolute Value Expression

Before solving an absolute value inequality, isolate the absolute value term on one side of the inequality. This step simplifies the problem and allows you to apply the rules for solving absolute value inequalities correctly.

Solving Linear Inequalities

After splitting the absolute value inequality into two linear inequalities, solve each by isolating the variable. This involves standard algebraic techniques such as adding, subtracting, multiplying, or dividing both sides while considering inequality direction.