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Variable isolation is the initial step in solving inequalities or equations where our goal is to get the variable on one side of the inequality sign by itself. This ensures we can easily see the range of values that the variable can take. For example, in the problem \( \frac{14y}{-3} \geq -18 \), the first thing we need to do is remove any fractions or negative signs attached to the variable.
To do this, multiply both sides by the same number that will clear the fraction or the term interfering with our variable. This helps keep the equation or inequality balanced and maintains its validity. In this specific example, multiplying by \(-3\) is necessary to eliminate the denominator, thereby isolating \(14y\).
Remember: when you manipulate an inequality by multiplying or dividing across it by a negative number, it's crucial to flip the inequality sign to preserve the truth of the inequality.

Whenever you are solving inequalities, a very important rule to remember is to flip the direction of the inequality sign when multiplying or dividing by negative numbers.
For instance, in our exercise, once we decide to multiply by \(-3\) to isolate the variable term \(14y\), we must turn the \(\geq\) into \(\leq\) as a part of the operation. This occurs because negative numbers invert the order on the number line.
Why does this happen? Consider numbers \(a < b\). If both sides are multiplied by \(-1\), the order reverses, because negative values closer to zero are indeed larger (less negative) than those further. Therefore, the inequality relationship between negative numbers hinges on this reversal.

Once you've isolated the variable in an inequality or equation, sometimes what you're left with includes fractions that need simplification for the best answer. Not only does this make your final answer neater, but it is also easier to interpret or use.
In our example, after isolating the variable \(y\), we end up with \( y \leq \frac{54}{14} \). Fractions like these are simplified by finding the greatest common divisor (GCD) of the numerator and the denominator, then dividing both by this number.
For \(54\) and \(14\), the GCD is \(2\), simplifying the fraction to \( \frac{27}{7} \). Simplifying helps in representing the inequality value in its simplest terms, making it clearer and more accurate for any subsequent calculations.