Solve each equation or inequality. | 6- 3x | 4 < -11

Verified step by step guidance

1. First, recognize that the given expression involves an absolute value inequality: \(|6 - 3x| \cdot 4 < -11\).
2. Divide both sides of the inequality by 4 to isolate the absolute value expression: \(|6 - 3x| < \frac{-11}{4}\).
3. Notice that the inequality \(|6 - 3x| < \frac{-11}{4}\) is not possible because the absolute value of any expression is always non-negative, meaning it cannot be less than a negative number.
4. Conclude that there are no solutions to this inequality because the condition is impossible to satisfy.
5. Therefore, the solution set is empty, indicating that no real number \(x\) will satisfy the given inequality.

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

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They use symbols like <, >, ≤, and ≥ to indicate whether one side is less than, greater than, or equal to the other. 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.

Case Analysis

When dealing with absolute value equations or inequalities, case analysis is a method used to break down the problem into simpler parts. For an expression like |A| < B, you would consider two cases: A < B and A > -B. This approach allows for a comprehensive solution by addressing all possible scenarios that satisfy the original condition.