Problem 59 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality. Then graph the solution set on the real number line. \(|x-20| \leq 4\)

Step-by-Step Solution

Answer

The solution to the inequality \(|x-20| \leq 4\) is \(16 \leq x \leq 24.\) This is represented on a number line by shading the region between and including the points representing 16 and 24.

Step 1: Isolate the absolute value expression

This inequality is already formatted correctly, with the absolute value expression by itself on the left side of the inequality: \(|x-20| \leq 4.\)

Step 2: Apply the definition of absolute value

An absolute value inequality \(|A| \leq B\) can be converted into a compound inequality of the form \(-B \leq A \leq B.\) Applying that here, we get \( -4 \leq x-20 \leq 4.\)

Step 3: Solve the compound inequality

We need to get 'x' alone on its side of each inequality. We do so by adding 20 to all parts of the compound inequality, resulting in \(16 \leq x \leq 24\).

Step 4: Graph the solution set on the real number line

The solution set includes all real numbers 'x' that satisfy the inequality (including 16 and 24). On a number line, we would place closed circles at the points representing 16 and 24 - indicating that these values are included in the solution set - and shade the region between them.

Key Concepts

Compound Inequalities

Compound inequalities are a key concept when dealing with absolute value inequalities. They are called "compound" because they consist of two separate inequalities connected by the word "and" or "or." In our exercise, the absolute value inequality is \( |x - 20| \, \leq 4 \), which can be rewritten as a compound inequality: \(-4 \leq x - 20 \leq 4\). Here, both inequalities need to be satisfied simultaneously, which is why they are connected by "and."

• The left part \(-4 \leq x - 20\) and the right part \(x - 20 \leq 4\) must both hold true.
• To solve, we aim to isolate the variable "x" by performing algebraic operations, such as addition or subtraction, on each part of the compound inequality.

Understanding compound inequalities is crucial, as they allow you to express a range of values that satisfy the inequality, instead of a single number.

Real Number Line

The real number line is a visual tool used in mathematics to represent continuous real number values. It extends infinitely in both directions, showing every possible real number sequentially from negative to positive infinity. In the context of inequalities, especially when dealing with solution sets, it becomes a useful means to visually depict where potential solutions lie. For the inequality \( 16 \leq x \leq 24\), the solution is graphed as follows:

• Position closed circles at 16 and 24 on the number line to signify that these endpoints are included in the solution set.
• Shade the section of the number line between these points to indicate that all numbers between 16 and 24 satisfy the inequality.

This approach not only provides a clear understanding of the range of solutions but also reinforces conceptual comprehension by merging visual and analytical reasoning.

Solving Inequalities

Solving inequalities involves determining the values of a variable that make the inequality true. Here's how we solved the exercise:

• We started with the compound inequality \(-4 \leq x - 20 \leq 4\), derived from the original absolute value inequality.
• To isolate "x," we added 20 to each part of the inequality, yielding \( 16 \leq x \leq 24 \).
• This means any value for "x" between 16 and 24, inclusive, will satisfy the initial inequality.

Key steps for solving inequalities include adding, subtracting, multiplying, or dividing each part of the inequality, similar to solving equations. However, it's important to remember that multiplying or dividing an inequality by a negative number reverses the inequality symbol. This particular rule does not apply to our exercise as we dealt only with positive numbers. By mastering these steps, you will be equipped to tackle more complex inequalities confidently.