91Ó°ÊÓ

To solve inequalities, the first step is often to isolate the variable term. This means you want to rearrange the inequality so that the term with the variable is by itself on one side. When dealing with expressions like \(-3x - 1 \leq 35\), you begin by eliminating other numbers on the side of the variable term. In this case, add 1 to both sides:- On the left, \(-3x - 1 + 1\) becomes \(-3x\).- On the right, \(35 + 1\) simplifies to \(36\).Now, the inequality reads \(-3x \leq 36\).Isolating the variable term sets the stage for solving the inequality by simplifying the equation structure, making it easier to isolate \(x\). This step is crucial because it reduces the complexity of the problem, allowing for easier manipulation and solving.

A significant concept in solving inequalities involves reversing the inequality sign. This rule is crucial when you multiply or divide both sides of an inequality by a negative number. Ignoring it can lead to incorrect solutions.In our example \(-3x \leq 36\), we must solve for \(x\) by dividing both sides by \(-3\). This division by a negative requires reversing the inequality sign:\[x \geq \frac{36}{-3}\]Why does the sign reverse? The change is rooted in the fundamental properties of inequalities. Multiplying or dividing by a negative number "flips" or changes the direction of the inequality.This is equivalent to reflecting the graph of the inequality across the origin on a number line, representing a switch in direction of the inequality's solution set.

Negative division plays a pivotal role in inequality problems. It adds complexity due to the special rule of reversing the inequality sign. In mathematical language, whenever you divide an inequality by a negative number, you must carefully take into account the change in direction of the inequality.In our example: Solving \(-3x \leq 36\) involves dividing by \(-3\) to make the coefficient of \(x\) equal to 1:- The left side \(-3x \div -3\) becomes\(x\).- The right side \(36 \div -3\) simplifies to \(-12\).After performing this step, don't forget to reverse the inequality sign, leading to \(x \geq -12\).When solving inequalities, always be mindful of operations involving negative numbers. Understanding this principle ensures your inequality solutions are accurate and reflective of the correct relationships between numbers.