Solve $|3x| + 3 < 2$
A) All real numbers
B) There is no solution.
C) $-1
D) $x>1$ or $x<-1$
Answer (1)
Isolate the absolute value term: ∣3 x ∣ < − 1 .
Recognize that the absolute value of any expression is non-negative.
Since ∣3 x ∣ cannot be less than -1, there is no solution.
The inequality has no solution: There is no solution.
Explanation
Understanding the Problem We are given the inequality ∣3 x ∣ + 3 < 2 . Our goal is to find the values of x that satisfy this inequality.
Isolating the Absolute Value First, we need to isolate the absolute value term. We can do this by subtracting 3 from both sides of the inequality: ∣3 x ∣ + 3 − 3 < 2 − 3 ∣3 x ∣ < − 1
Analyzing the Inequality Now, let's analyze the inequality. The absolute value of any real number is always non-negative (i.e., greater than or equal to 0). In other words, ∣3 x ∣ ≥ 0 for all real numbers x .
Conclusion Since ∣3 x ∣ is always greater than or equal to 0, it can never be less than -1. Therefore, there is no value of x that can satisfy the inequality ∣3 x ∣ < − 1 .
Final Answer Thus, the inequality ∣3 x ∣ + 3 < 2 has no solution.
Examples
Absolute value inequalities can be used in various real-life scenarios, such as determining the acceptable range of error in manufacturing processes. For example, if a machine is designed to produce bolts with a diameter of 10mm, an absolute value inequality can be used to define the tolerance range. If the tolerance is 0.1mm, the inequality would be ∣ d − 10∣ < 0.1 , where d is the actual diameter of the bolt. This ensures that the bolts produced are within the acceptable size range. In finance, absolute value inequalities can be used to model risk. For example, an investor might want to limit the daily fluctuation of an investment to a certain percentage. If the target is a 2% fluctuation, the inequality would be ∣ ( V t − V t − 1 ) / V t − 1 ∣ < 0.02 , where V t is the value of the investment on day t .
