91Ó°ÊÓ

When solving inequalities, the goal is to find a range of values for a variable that satisfy the inequality condition. Inequalities are like equations but instead of an equal sign, they use symbols such as greater than ">", less than "<", and their non-strict variations ">=" and "<=". For instance, in the inequality given by the expression \(1 < 2x - 4 \leq 7\), we're looking to determine the values of \(x\) that satisfy this condition simultaneously in both parts of the inequality. To solve an inequality:

• First isolate the variable by performing arithmetic operations such as adding, subtracting, multiplying, or dividing both sides of the inequality.
• If you multiply or divide by a negative number, you must flip the inequality sign.
• Splitting complex inequalities into simpler ones, solving each, and then combining the solutions can help simplify the process.

In our example, the inequality is split into two parts: \(1 < 2x - 4\) and \(2x - 4 \leq 7\). Solving these gives us the range \(\frac{5}{2} < x \leq \frac{11}{2}\). This means any \(x\) more than \(\frac{5}{2}\) and up to \(\frac{11}{2}\) satisfies the condition.

Set notation is a way of specifying a collection of objects or numbers, often to describe the solution to equations or inequalities. Sets are usually enclosed in curly braces \(\{ \}\) and list the included elements clearly. In this exercise, the set \(S\) is written as \(S = \{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\), showcasing the elements under consideration.Using set notation is helpful when:

• We want to clearly define which elements are part of our discussion or solution.
• Express solutions concisely, especially in mathematical problems involving multiple possible values.
• Compare elements against conditions or ranges effectively.

In our exercise, we use set notation to state which elements of \(S\) satisfy the inequality \(\frac{5}{2} < x \leq \frac{11}{2}\). It helps narrow down from a larger list of elements to those that meet specific criteria.

Element comparison in sets is the method of checking each element of a set against a specific condition or range and determining which satisfy it. For the inequality \(\frac{5}{2} < x \leq \frac{11}{2}\), we take each element from the original set \(S = \{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\) and see whether it falls within this specific range.Here’s how you compare elements in a set:

• Take each element in the set and plug it into the inequality to see if the condition holds.
• Only include the elements that meet the inequality criteria in the final solution set.

In this exercise, after evaluating each element, only \(4\) falls within the specified range \(\left(\frac{5}{2}, \frac{11}{2}\right]\). This systematic comparison ensures that we accurately determine which elements fulfill the conditions of the inequality.