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

Verified step by step guidance

1. The given inequality is |3x - 7| + 1 < -2. This involves an absolute value expression, which is always non-negative. Therefore, the expression |3x - 7| + 1 cannot be less than -2, as the smallest value it can take is 1 (when |3x - 7| = 0).
2. Since the left side of the inequality, |3x - 7| + 1, is always greater than or equal to 1, it can never be less than -2. This means there are no real values of x that satisfy the inequality.
3. Since the inequality |3x - 7| + 1 < -2 has no solution, we conclude that there are no values of x that make the inequality true.

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 and inequalities, absolute values can create two separate cases to consider, as they can be either positive or negative.

Inequalities

Inequalities express a relationship where one side is not equal to the other, using symbols like <, >, ≤, or ≥. They indicate that one quantity is less than or greater than another. Solving inequalities often involves similar steps to solving equations, but requires careful consideration of the direction of the inequality when multiplying or dividing by negative numbers.

Solving Compound Inequalities

Compound inequalities involve two or more inequalities that are connected by 'and' or 'or'. To solve them, one must find the values that satisfy all parts of the inequality. In the context of absolute value inequalities, this often leads to breaking the problem into two separate cases, allowing for a comprehensive solution that encompasses all possible scenarios.