91Ó°ÊÓ

In equations and inequalities alike, isolating the variable is a fundamental step towards finding the solution. This involves moving all terms containing the variable to one side and all other terms to the opposite side. To accomplish this, you can add, subtract, multiply, or divide both sides of the equation or inequality by the same number, as long as the number isn't zero.

For example, consider the inequality \( -\frac{1}{3} y + 5 < 7 \). To isolate \(y\), we must move the constant term, 5, to the opposite side. By subtracting 5 from both sides, as performed in Step 1 of the solution, we advance towards a simpler form \(-\frac{1}{3} y < 2\), with \(-\frac{1}{3} y\) isolated on one side.

When you are working to solve an inequality, you manipulate the terms and operations in order to isolate the variable and find its range of solutions. Inequality manipulation follows many of the same rules as solving equations. However, there is one crucial difference: if you multiply or divide an inequality by a negative number, you must flip the inequality sign.

This rule ensures that the inequality still holds true. For instance, if we have \(-\frac{1}{3} y < 2\) and multiply both sides by -3, the inequality changes direction, resulting in \(y > -6\). This step, as seen in Step 2 of our exercise, is vital for retaining the integrity of the original inequality.

Dealing with negative coefficients in inequalities requires careful attention to ensure the solution remains accurate. As per the rules, if the coefficient is negative, any effort to eliminate it, specifically through multiplication or division, leads to the reversal of the inequality sign.

In the given exercise, the coefficient of \(-\frac{1}{3} y\) is negative. In the process of solving for \(y\), multiplying both sides by -3, the reciprocal of the coefficient, negates the coefficient and thus flips the inequality from '<' to '>'. This is why we end up with \(y > -6\) as the solution. Understanding this rule is critical since neglecting it can lead to an incorrect solution and possibly a misunderstanding of the variable's feasible value range.