Solve each equation or inequality. | 4.3x + 9.8| < 0

Verified step by step guidance

1. Step 1: Understand the nature of absolute value inequalities. The expression \(|4.3x + 9.8| < 0\) implies that the absolute value of \(4.3x + 9.8\) is less than zero.
2. Step 2: Recall that the absolute value of any real number is always non-negative, meaning it is either zero or positive.
3. Step 3: Since an absolute value cannot be negative, the inequality \(|4.3x + 9.8| < 0\) has no solution.
4. Step 4: Conclude that there are no real numbers \(x\) that satisfy the inequality.
5. Step 5: Therefore, the solution set is empty, often denoted as \(\emptyset\) or \(\{\} \).

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving equations and inequalities that involve it.

Inequalities

Inequalities express a relationship where one quantity is less than, greater than, or not equal to another. In this case, the inequality |4.3x + 9.8| < 0 suggests that the expression inside the absolute value must be less than zero. However, since absolute values cannot be negative, this inequality has no solution, highlighting the importance of recognizing the properties of inequalities.

Solution Sets

A solution set is the collection of all values that satisfy a given equation or inequality. In the context of the provided inequality, understanding that the absolute value cannot be negative leads to the conclusion that there are no values of x that satisfy |4.3x + 9.8| < 0. This concept is essential for determining the validity of solutions in algebraic expressions.