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

Verified step by step guidance

1. Step 1: Start by isolating the absolute value expression. Add 2 to both sides of the inequality: \(|5x + \frac{1}{2}| - 2 + 2 < 5 + 2\).
2. Step 2: Simplify the inequality: \(|5x + \frac{1}{2}| < 7\).
3. Step 3: Set up two separate inequalities to solve for \(x\): \(5x + \frac{1}{2} < 7\) and \(5x + \frac{1}{2} > -7\).
4. Step 4: Solve the first inequality \(5x + \frac{1}{2} < 7\) by subtracting \(\frac{1}{2}\) from both sides and then dividing by 5.
5. Step 5: Solve the second inequality \(5x + \frac{1}{2} > -7\) by subtracting \(\frac{1}{2}\) from both sides and then dividing by 5.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For example, |x| equals x if x is positive and -x if x is negative. In the context of inequalities, understanding how to manipulate absolute values is crucial for solving equations that involve them.

Inequalities

Inequalities express a relationship where one side is not equal to the other, using symbols like <, >, ≤, or ≥. Solving inequalities often involves isolating the variable, similar to equations, but requires careful consideration of the direction of the inequality when multiplying or dividing by negative numbers.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. This process typically includes combining like terms, isolating the variable, and applying inverse operations. In the case of inequalities, the same principles apply, but the solution set may include a range of values rather than a single point.