Problem 84 - College Algebra

Question

Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$\left|2-\frac{x}{2}\right|-1 \leq 1$$

Step-by-Step Solution

Answer

The solution to the inequality is \( 2 \leq x \leq 8 \), graphed on a number line with a closed interval from 2 to 8, and expressed in interval notation as [2, 8].

Step 1: Rewrite the absolute value inequality

First, rewrite the given absolute value inequality \(\left|2-\frac{x}{2}\right|-1 \leq 1\) as \( \left|2-\frac{x}{2}\right|\leq 2\). This is achieved by adding 1 to both sides.

Step 2: Formulate equivalent inequalities

Next, formulate two equivalent inequalities without absolute value according to the property of absolute values. This gives \( -2 \leq 2-\frac{x}{2} \leq 2 \).

Step 3: Solve each inequality for x

Combine the inequalities into a single one for easier solution. Convert the compound inequality \( -2 \leq 2-\frac{x}{2} \leq 2 \) into its equivalent form \( 2 \leq x \leq 8 \).

Step 4: Graph the solution on a number line

Draw a number line and mark x at every integer. The solutions to the inequality include x such that \( 2 \leq x \leq 8 \), inclusive. Therefore, mark a closed circle at x = 2 and x = 8 on the number line, and shade the interval between them.

Step 5: Express the solution using interval notation

The final step is to write the answer in interval notation. For \( 2 \leq x \leq 8 \), the interval notation is [2, 8]. This indicates that every number between 2 and 8, including 2 and 8 themselves, is a solution to the original inequality.

Key Concepts

Understanding Inequalities

Inequalities are mathematical statements indicating that one quantity is less than or greater than another. Often represented by symbols like <, >, ≤, and ≥, they show the relationship between two expressions. In our exercise, we dealt with an inequality involving absolute values. An absolute value inequality like \(|a| \leq b\) implies that the distance from a to zero on the number line is no more than b. This leads us to two simple inequalities: \

\
• The expression inside the absolute value is less than or equal to the positive b.
\
• The expression inside the absolute value is greater than or equal to the negative b.
\

\Once you write these expressions, it’s essential to solve them to find the range or set of solutions that satisfy both conditions. Understanding this is crucial for correctly graphing and noting solutions.

Interval Notation Explained

Interval notation is a concise way of describing sets of numbers, particularly those that occur on a number line. It uses brackets to show which numbers are included or excluded. In our exercise, we encountered the interval \([2, 8]\). Here’s how to read interval notation:

• Square Brackets [ ]: means the number is included in the set. This is used when the inequality is ≤ or ≥.
• Parentheses ( ): means the number is not included. This is used for < or >.

For instance, in \([2, 8]\), both 2 and 8 are included in the solution set. This shows all possible values between 2 and 8 that satisfy the inequality condition.

Graphing on a Number Line

Graphing inequalities on a number line is a simple visual way to understand the solutions. It involves a few steps to ensure clarity. Let’s break it down:

• Draw the Line: Start by drawing a straight horizontal line and marking the relevant numbers, such as the integers around the solution interval.
• Mark Points: Use closed circles (●) if the endpoint is included (like in ≤ or ≥), and open circles (○) if it is not (like in < or >).
• Shade the Region: Once the points are marked, shade the region between them. This shading represents all the numbers that satisfy the inequality.

For the exercise, we marked at 2 and 8, shading between them to indicate the set \([2, 8]\). This visual representation helps to quickly understand and check solution ranges.