Problem 37 - College Algebra

Question

For the following exercises, graph the function. Observe the points of intersection and shade the \(x\) -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. $$ |x-2|<0 $$

Step-by-Step Solution

Answer

No solution exists for \(|x-2| < 0\).

Step 1: Understanding the Absolute Value Inequality

The inequality given is \(|x-2| < 0\). Absolute values, by definition, are always non-negative. This means that they are either positive or zero. Therefore, it is impossible for an absolute value to be less than zero. Such an inequality has no solution.

Step 2: Graphing the Function

Since there is no solution to the inequality \(|x-2| < 0\), there are no points to graph on the coordinate plane. The absolute value function \(y = |x-2|\) intersects the x-axis at \(x=2\), but there is no x-axis shading since the inequality cannot be satisfied.

Step 3: Interval Notation for the Solution Set

Given that \(|x-2| < 0\) has no solution, there is no interval notation to represent a solution set. The correct representation is to denote the empty set, which can be written as \(\emptyset\) or as an empty interval \(()\).

Key Concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics that assigns the distance of a number from zero on the number line without considering the direction. For example, both -4 and 4 have an absolute value of 4 since they are both four units away from zero. In mathematical terms, the absolute value of a number \( a \) is represented as \( |a| \).
The absolute value function is quite handy in real-world situations where only the size of a value matters and not whether it is positive or negative. This function is always non-negative.

• For positive numbers, \( |a| = a \)
• For zero, \( |0| = 0 \)
• For negative numbers, \( |a| = -a \)

Understanding this makes absolute value inequalities, such as \( |x-2|<0 \), interesting since an absolute value can never be negative. This inequality will thus have no solution because absolute values are never less than zero.

Graphing Inequalities

Graphing inequalities can help you visually see which parts of the number line make the inequality true. In the exercise, we look at \( |x-2| < 0 \), which suggests a challenging situation since we know absolute values can't be negative. When we graph typical inequalities, we highlight the solution values to visualize which parts of the number line are included in the solution.
For an absolute value inequality, we normally check:

• If the inequality includes solutions on both sides of the origin, or
• It opens to one side only.

In this case, since \( |x-2| < 0 \) gives us no solution, there are no sections of the number line to color or shade, emphasizing the lack of solutions.

Interval Notation

Interval notation is a way to describe sets of numbers between endpoints on the number line. It gives a concise method to express which parts of the number line satisfy an inequality. Using parentheses and brackets, we can easily communicate ranges of numbers.
- A bracket \( [ ] \) includes the endpoint.- A parenthesis \( () \) does not include the endpoint.
For example:

• \( (1, 5) \) describes all numbers greater than 1 and less than 5.
• \( [1, 5] \) includes both 1 and 5 as solutions.

When dealing with no solutions, as in \( |x-2| < 0 \), we express this using the empty set notation \( \emptyset \) or by writing an empty interval \( () \), indicating there are no numbers in the solution set.

No Solution Inequalities

Sometimes, inequalities such as \( |x-2| < 0 \) indicate that no possible solution exists. Such inequalities are aptly named 'no solution inequalities'. These occur when the conditions of the inequality cannot logically be met.
Understanding why an inequality has no solution is crucial:

• Absolute value is inherently non-negative.
• An expression like \( |x| < 0 \) implies a negative value which is impossible.
• Therefore, such inequalities don’t produce any valid numbers within their constraints.

Thus, the solution set is empty, expressed as the empty set \( \emptyset \) or using empty interval notation \( () \). This notation effectively communicates that no values satisfy the inequality's condition.