Problem 23 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ |x-5| \leq 3 $$

Step-by-Step Solution

Answer

The solution is \([2, 8]\).

Step 1: Understand the Absolute Value Inequality

The inequality given is \(|x-5| \leq 3\). This means that the distance between \(x\) and 5 on a number line is less than or equal to 3. We can break this inequality into two separate conditions to find the values of \(x\).

Step 2: Split the Absolute Inequality into Two Inequalities

The inequality \(|x-5| \leq 3\) can be rewritten as two inequalities: 1. \(x - 5 \leq 3\)2. \(x - 5 \geq -3\)These represent the maximum and minimum distances from 5.

Step 3: Solve the First Inequality

Solve \(x - 5 \leq 3\) by adding 5 to both sides:\[x \leq 8\]

Step 4: Solve the Second Inequality

Solve \(x - 5 \geq -3\) by adding 5 to both sides:\[x \geq 2\]

Step 5: Combine Solutions

The solutions \(x \leq 8\) and \(x \geq 2\) can be combined to show that \(x\) is between 2 and 8, inclusive. Thus, the solution to the inequality is an interval.

Step 6: Express in Interval Notation

The solution set can be expressed in interval notation as \([2, 8]\), which includes all numbers from 2 to 8, inclusive.

Key Concepts

Interval Notation

Interval notation is a mathematical way of representing sets of numbers along a number line. It helps to define the subset of numbers for which an inequality holds true. In the given exercise, the inequality solution is expressed using interval notation.To express our solution of \( |x-5| \leq 3 \) in interval notation, we found from our steps that the variable \( x \) is between 2 and 8. This indicates that the values that satisfy the inequality range from 2 to 8, inclusive.

• The square brackets \( [ ] \) indicate that the endpoints are included in the solution.
• Hence, our final interval notation is \( [2, 8] \), signifying that every number from 2 to 8 including 2 and 8 itself falls within the solution set.

Interval notation is very useful as it provides a clear, concise representation of the values satisfying an inequality, making it easier to understand and communicate the solution.

Solving Inequalities

Solving inequalities involving absolute values can initially seem tricky, but by understanding the concept, it becomes straightforward. In this case, the expression \( |x-5| \leq 3 \) suggests that the distance of \( x \) from 5 should be at most 3.The main idea is to break down the absolute value inequality into two separate inequalities:

• Subtract the constant from the absolute expression: \( x - 5 \leq 3 \), meaning \( x \) can be at most 3 units to the right of 5.
• Also, \( x - 5 \geq -3 \), meaning \( x \) can be at most 3 units to the left of 5.

By solving these individual inequalities, we ultimately find that \( x \leq 8 \) and \( x \geq 2 \), giving us a range of solutions. By understanding these steps, students will get better at identifying how the absolute boundary approach helps to pinpoint the values \( x \) can take.

Number Line Representation

Using a number line representation helps students visualize and better understand the solutions to inequalities.For our solution \( [2, 8] \), we can represent it on a number line to see which numbers satisfy \( |x-5| \leq 3 \). Here’s how:

• Draw a horizontal line and place numbers on it from 0 to 10 for convenience.
• Locate the numbers 2 and 8, marking them with solid dots. The solid dots indicate that these endpoints are included in the function as the inequality includes the endpoints \( (\leq, \geq) \).
• Shade the region between these points, from 2 to 8, to show that any number in this range is a valid solution.

This visual aid not only reaffirms the solution but also assists learners in comprehensively grasping concepts tied to inequalities and interval solutions. A number line provides a clear picture of how all valid values configure within the given bounds.