Solve each equation or inequality. | 3x- 7 | + 1 > -2

Verified step by step guidance

1. Start by examining the inequality: \(| 3x - 7 | + 1 < -2\). The absolute value expression \(| 3x - 7 |\) is always greater than or equal to zero, so consider the implications of the inequality involving a negative number on the right side.
2. Subtract 1 from both sides to isolate the absolute value term: \(| 3x - 7 | < -2 - 1\), which simplifies to \(| 3x - 7 | < -3\).
3. Recall that the absolute value of any real number is always non-negative (i.e., \(|a| \geq 0\) for any \(a\)). Therefore, it is impossible for \(| 3x - 7 |\) to be less than a negative number like \(-3\).
4. Since the inequality \(| 3x - 7 | < -3\) has no solution, conclude that the original inequality \(| 3x - 7 | + 1 < -2\) also has no solution.
5. Summarize that there are no real values of \(x\) that satisfy the inequality because the absolute value expression cannot be less than a negative number.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of a variable or expression is compared to a number. Understanding how to interpret and solve inequalities like |A| < B or |A| > B is essential, noting that absolute values are always non-negative.

Properties of Absolute Value

The absolute value of a number represents its distance from zero on the number line and is always non-negative. This means expressions like |3x - 7| cannot be negative, which affects the possible solutions when combined with inequalities.

Solving Inequalities

Solving inequalities requires isolating the variable and considering the inequality direction. When dealing with absolute values, it is important to analyze whether the inequality is possible or has no solution, especially if the inequality contradicts the non-negativity of absolute values.