91Ó°ÊÓ

Interval notation is a concise way to represent a set of numbers on the number line. When you solve inequalities, you'll often use interval notation to express the solution set. In our example, after solving the linear inequality, we found that the solution for \( x \) was \( 7.5 < x \leq 10.5 \).
This inequality tells us that \( x \) can be any number greater than 7.5 and up to 10.5, including 10.5 itself. In interval notation, this is expressed as

• ``(7.5, 10.5]``.

The parenthesis ``(`` and brackets ``[`` in interval notation have a particular meaning:

• A parenthesis ``(`` indicates that the number is not included in the interval, also known as "open."
• A bracket ``[`` indicates that the number is included in the interval, referred to as "closed."

So, in

• ``(7.5, 10.5]``

the value 7.5 is not included in the solution (open), while 10.5 is included (closed). This is an easy, uniform method to display continuous values from one number to another. Using interval notation helps in quickly identifying and understanding the solution range.

Graphing the solution set of an inequality provides a visual representation that can make understanding the solution easier. For the inequality \( 7.5 < x \leq 10.5 \), graphing helps us see the part of the number line that satisfies this condition. Here’s how you can graph it:
First, you place a number line and mark the points 7.5 and 10.5 on it. Then:

• At 7.5, you draw an open circle, indicating that this number is not included in the solution set. Open circles represent values that are not part of the solution.
• At 10.5, you draw a closed circle, signifying that this number is included in the solution set. Closed circles mean the value is included.

Next, you shade the region between 7.5 and 10.5 to show all the possible solutions for \( x \). The shading indicates every number that makes the inequality true. Looking at the graph tells you the solution at a glance, which is especially useful for understanding or explaining math problems visually.

Solving a linear inequality involves a series of straightforward steps aimed at isolating the variable of interest to find a solution range. The inequality given was \( \frac{1}{6} < \frac{2x-13}{12} \leq \frac{2}{3} \). Let's break down the process:
**Step 1: Eliminate Fractions**

• Fractions can complicate calculations, so it's often helpful to remove them early. In this problem, multiplying through by 12 did the trick, simplifying the inequality to \( 2 < 2x - 13 \leq 8 \).

**Step 2: Isolate the Variable**

• To focus on the variable \( x \), isolate it by performing inverse operations. Add 13 to all parts, resulting in \( 15 < 2x \leq 21 \).

**Step 3: Solve for \( x \)**

• The final step involves solving for \( x \) by dividing the entire inequality by 2, reaching \( 7.5 < x \leq 10.5 \).

Each step employs basic algebraic principles. Properly following them guarantees that you'll find the correct range of values that \( x \) can assume, ensuring the inequality holds.