91Ó°ÊÓ

Solving inequalities requires a systematic approach to ensure clarity and accuracy. Let's break down the steps involved in solving the inequality given in the exercise: - Start by rearranging the inequality so that all the terms containing the variable are on one side, while the constants are on the other. For our inequality \( -2s - 105 \leq -7s - 205 \), we add \(-7s\) to both sides to collect the variable terms together: \[-2s + 7s - 105 \leq -205\]- Next, simplify the expression: combine the \(s\) terms. This simplifies to: \[5s - 105 \leq -205\]- Isolating the variable term involves removing the constant from the side with the variable. Add 105 to both sides: \[5s \leq -205 + 105\] This results in: \[5s \leq -100\]- Finally, solve for \(s\) by dividing both sides by 5: \[s \leq \frac{-100}{5}\] Simplifying gives: \[s \leq -20\]These steps help in isolating the variable \(s\) effectively and finding the solution of the inequality.

Graphing an inequality on a number line visually represents the solution set, making it easier to understand. After solving the inequality \(s \leq -20\), the next step is to graph this solution. - Begin by drawing a horizontal line to represent the number line.- Locate the point \(-20\) and mark it clearly on the number line.- Since the inequality \(s \leq -20\) includes \(-20\), draw a filled-in circle at \(-20\) on the number line to represent that \(-20\) is part of the solution.- Shade all the numbers to the left of \(-20\). This shows all the values \(s\) can take under the inequality, meaning all numbers less than \(-20\) are included in the solution set. Graphing provides a clear and intuitive way to see which numbers satisfy the inequality.

Interval notation is a concise way of writing the set of solutions for an inequality. This notation efficiently communicates the range of values that satisfy the inequality. For our solution \(s \leq -20\), it is important to understand how to express this using interval notation. - Since \(-20\) is the largest number in the solution set and is included, we write the right bracket as a square bracket: \([-20]\).- The solution includes all numbers less than \(-20\), extending infinitely to the left on the number line. Infinity is written in interval notation with a parenthesis, which indicates that infinity is never a closed set since it isn't a specific number. Thus, we use \((-\infty, -20]\). Utilizing interval notation conveys that \(s\) can be any number from negative infinity up to and including \(-20\). This method is not only succinct but also widely understood in mathematics, providing clarity in the communication of solution sets.