91Ó°ÊÓ

Set notation is a concise and systematic way to represent a collection of objects, usually numbers. Sets are denoted by listing each element inside curly braces (\( \{ \} \)). For example, in our exercise, the set \(S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\) includes eight elements. Each element is separated by a comma.

Sets can include:

• Integers like \(-2\) and \(1\).
• Fractions like \(\frac{1}{2}\).
• Irrational numbers such as \(\sqrt{2}\), which cannot be expressed as a simple fraction.

Understanding set notation is crucial when solving inequalities, as it helps you easily identify which values need to be checked against the inequality. Always pay attention to ensure the values within the set satisfy any given conditions. This approach offers clarity and precision in identifying the suitable candidates from a collection of numbers for testing inequalities.

Solving inequalities involves finding the range of values for a variable that satisfies a given condition. In our problem, we dealt with the compound inequality \(1 < 2x - 4 \leq 7\). Let's break down the process:

1. **Part 1: Simplify the Inequality** - Start with the inequality \(1 < 2x - 4\). - Add 4 to each side to get \(5 < 2x\). - Divide each side by 2, resulting in \(x > 2.5\).

2. **Part 2: Simplify the Other Side** - Now consider \(2x - 4 \leq 7\). - Add 4 to both sides, giving \(2x \leq 11\). - Divide by 2, which simplifies to \(x \leq 5.5\).

3. **Combine Results** - Combining part 1 and part 2, we determine that \(2.5 < x \leq 5.5\).

By following these steps methodically, you can solve similar inequalities. Remember, when you divide or multiply by a negative number, the inequality sign flips direction. This rule is critical when working with inequalities involving negative coefficients.

Algebraic expressions play a key role in solving inequalities. These expressions typically involve variables, numbers, and arithmetic operations. In this exercise, \(2x - 4\) is the algebraic expression central to the inequality \(1 < 2x - 4 \leq 7\).

Understanding each part of an algebraic expression helps you effectively manipulate and solve for the variable.
- **Variables**: Symbols like \(x\) represent unknown quantities that we aim to find.- **Constants**: Numbers such as \(-4\) are constants since they do not change.- **Operations**: Include addition, multiplication, etc., which allow you to rearrange and simplify expressions.
In our exercise, applying algebraic methods to rearrange \(2x - 4\) helped isolate \(x\), allowing us to find values that satisfy the original inequality. It's all about understanding how to treat the variable 'x' and systematically apply operations to solve real-world mathematical problems.