Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Solve these inequalities and explain your answers: Think carefully. (a) \(|3 x-4|>-4\) (b) \(|3 x-4|>0\) (c) \(|3 x-4|<-4\)

Step-by-Step Solution

Answer

(a) The solution set is all real numbers. (b) The solution set is all real numbers except \(x = 4/3\). (c) There are no solutions.

Step 1: Solve inequality \(|3 x-4|>-4\)

Firstly, note that the absolute value of a number is always nonnegative. Therefore, any number is greater than -4, so this inequality holds true for all x values.

Step 2: Solve inequality \(|3 x-4|>0\)

The absolute value of a number is equal to 0 if and only if the number inside the absolute value sign is equal to 0. Here, \(3x - 4 = 0\). We solve for x to yield \(x = 4/3\). Therefore, \(|3x-4|\) is greater than 0 if and only if \(x ≠ 4/3\).

Step 3: Solve inequality \(|3 x-4|<-4\)

Similarly to the first inequality, the absolute value of a number is always nonnegative. So it can't be less than -4. Thus, no solutions exist for this inequality.

Key Concepts

Understanding Absolute Value

Absolute value is a mathematical concept that is used to describe the distance of a number from zero on a number line. This distance is always expressed as a nonnegative number, regardless of whether the original number is positive, negative, or zero itself.

The absolute value function is denoted by vertical bars, such as \( |x| \). If \( x \) is positive, \( |x| = x \). If \( x \) is negative, \( |x| = -x \). For instance, \( |-3| = 3 \) because \( -3 \) is 3 units away from zero on the number line.

• Absolute value converts all input to a nonnegative output.
• The only time \( |x| = 0 \) is when \( x = 0 \).

Understanding this concept is crucial for solving any problems involving absolute values and inequalities, as it sometimes leads to specific scenarios, such as those listed in the exercise above.

Solving Inequalities

Inequalities are mathematical expressions that compare two values using inequality signs like \( >, <, \geq, \), and \(< \).When working with absolute values in inequalities, it is important to consider the properties of absolute values. Let's look at some crucial points through the given inequalities:

• When solving \(|3x-4| > a\), consider two cases: either \(3x-4 > a\) or \(3x-4 < -a\).
• The inequality \(|3x-4| < a\) will generally have solutions only if \( a \) is a positive number.
• If \( |expression| = 0 \), it implies that the expression within the absolute value must be zero.

In problem (a), \(|3x-4| > -4\) is always true because the absolute value cannot be negative. In problem (b), think about when the expression inside would make the absolute value non-zero. Problem (c), \(|3x-4| < -4\), has no solution due to the absolute value being always zero or positive.

Exploring Nonnegative Numbers

A nonnegative number is any number that is either positive or zero. This is a fundamental principle that comes into play when dealing with absolute values. Since the absolute value of any number is nonnegative, these problems often check for situations where inequalities hold true or are impossible.

• Numbers that are \( \geq 0 \) include both positive numbers and zero.
• The key reason behind the truth of some inequalities with absolute values is this property of nonnegativity.

For example, in problem (a) \(|3x-4| > -4\), any absolute value expression satisfies this because it is impossible for the result to be less than \( -4 \). Meanwhile, in problem (c), nonnegative numbers can never satisfy \(|expression| < -4\), thus demonstrating a typical characteristic of such scenarios.