Problem 61 - Precalculus Student Solutions Manual 5th

Question

Sohe each equation or inequality. \(|4.3 x+9.8|<0\)

Step-by-Step Solution

Answer

No solution

Step 1: Understand the Absolute Value Property

An absolute value \( |A| \) represents the distance of \( A \) from zero on the number line. Thus, it is always non-negative. This means \( |A| < 0 \) is never possible because absolute values cannot be negative.

Step 2: Analyze the given Inequality

Examine the given inequality \( |4.3x + 9.8| < 0 \). Since an absolute value cannot be negative, the inequality \( |4.3x + 9.8| < 0 \) is not possible.

Step 3: Conclusion

Since the absolute value of any real number cannot be less than 0, the given inequality does not have any solution.

Key Concepts

Absolute Value Properties

The absolute value of a number is a measure of the number's distance from zero on the number line, regardless of direction. Because distance cannot be negative, absolute values are always non-negative. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5. This is written as: - | -5 | = 5 - | 5 | = 5 When solving inequalities involving absolute values, it's important to remember that the result of the absolute value function will always be greater than or equal to zero. Therefore, any inequality suggesting that an absolute value is less than zero is not possible and has no real solution.

Inequalities

Inequalities are mathematical statements that compare two values or expressions. They use symbols such as: - '<' (less than) - '<=' (less than or equal to) - '>' (greater than) - '>=' (greater than or equal to) When dealing with absolute value inequalities, there are two main types: - Absolute value less than ( |A| < B ): This means that A is between -B and B, excluding -B and B themselves. - Absolute value greater than ( |A| > B ): This signifies that A is greater than B or less than -B. For example, if we have |x| < 3, it implies -3 < x < 3. Conversely, |x| > 3 implies x < -3 or x > 3.

Number Line

The number line is a visual tool used in mathematics to represent numbers as points along a line. Each point on the line corresponds to a specific number, increasing from left to right. It can be used to illustrate absolute values and inequalities effectively. On a number line: - Zero (0) is positioned at the center. - Positive numbers are placed to the right of zero. - Negative numbers are positioned to the left of zero. When solving inequalities involving absolute values, the number line helps us to visualize the distances and positions of the numbers relative to zero. For example, if we want to solve |x| < 3 using a number line, we shade the region between -3 and 3, indicating that x can take any value in that range. Using a number line also helps make the concept of distance more intuitive and accessible.