Problem 52 - College Algebra

Question

Solve each inequality in Exercises 49-56 and graph the solution set on a number line. Express the solution set using interval notation. $$-6

Step-by-Step Solution

Answer

The solution to the inequality \( -6 < x - 4 \leq 1 \) is \( x \) is in the interval \( (-2, 5] \)

Step 1: Make inequality easier to read

Let's rewrite the inequality in a way that makes it easier to read: \( -6 < x - 4 \leq 1 \)

Step 2: Isolate 'x'

To isolate \( x \), we'll add 4 to all sides of the inequality. This gives us: \( -6 + 4 < x - 4 + 4 \leq 1 + 4 \), which simplifies to \( -2 < x \leq 5 \)

Step 3: Express solution in interval notation

Interval notation is a way of writing subsets of the real number line. A round bracket '(' or ')' means the endpoint is not included in the set, while a square bracket '[' or ']' means the endpoint is included. For our solution, \( x > -2 \) and \( x \leq 5 \), the interval notation would be \( (-2, 5] \)

Key Concepts

Interval Notation

Interval notation is a succinct and universally accepted method for describing intervals on the real number line. In mathematics, this form of notation is invaluable when we need to express the range of values for which an inequality holds true.

Let's delve into how this works with a concrete example. Consider the inequality \( -2 < x \leq 5 \). This statement asserts that the value of \( x \) is greater than -2 but less than or equal to 5. To express this in interval notation, we use a combination of parentheses and brackets. The parenthesis, \( ( \) or \( ) \), indicates that the endpoint is not included, meaning the set of numbers does not actually reach that boundary. A bracket, such as \( [ \) or \( ] \), means the endpoint is part of the set.

In this particular inequality, the interval starts just beyond -2 (as it is not included in the solution set due to the strict inequality <), and it goes up to and includes 5 (because of the 'less than or equal to' relationship, denoted by \leq). Thus, using interval notation, this inequality is represented as \( (-2, 5] \) - an elegant and concise way to depict the range of possible values for \( x \).

Number Line Graphing

Number line graphing is the visual representation of numerical values or ranges on a straight line, where each point corresponds to a particular number. This method is excellent for visualizing inequalities and their solutions.

In our example inequality, \( -2 < x \leq 5 \), graphing on a number line helps us see exactly which numbers satisfy the condition. On a number line, we would represent this by:

• Drawing a line that includes numbers from less than -2 to 5 and beyond.
• Putting an open circle, or sometimes a small gap, at -2, to show that this point is not included in the solution.
• Placing a closed dot or a filled-in circle at 5, to indicate that this endpoint is indeed part of the solution set.

The area of the number line between the -2 (not including it) and the 5 (including it) would typically be shaded or bolded to depict the range of acceptable solutions. This graphical representation offers an immediate visual comprehension of the range of values that \(x\) can assume.

Inequality Isolation

Inequality isolation is a crucial algebraic technique used to find the solution set to an inequality. The goal is to manipulate the inequality to express the variable of interest, typically \(x\), on one side of the inequality symbol, leaving a numerical value or a simpler expression on the other side.

As with the given textbook exercise, we start with the compound inequality \(-6 < x - 4 \leq 1\). The isolation process involves performing the same operation on each part of the inequality to keep the equation balanced. Since our aim is to solve for \(x\), we add 4 to all parts of the inequality:
\[ -6 + 4 < x - 4 + 4 \leq 1 + 4 \]
This simplifies to \(-2 < x \leq 5\), successfully isolating \(x\). It's important to maintain the direction of the inequality throughout the isolation process. If at any point we multiply or divide by a negative number, we must remember to reverse the inequality sign — a critical step that's often overlooked but essential for correct solutions. By mastering isolation, students are empowered to handle a wide range of inequalities efficiently and confidently.