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In solving inequalities, it is crucial to isolate the variable, which simplifies the task of finding the range of its possible values. Imagine the inequality as a simple balance scale. Our objective is to keep this balance by isolating the variable. This often involves moving other elements to the opposite side. For the example inequality given, \( 8 > \frac{q}{3} \), isolation requires us to first eliminate the fraction.

By multiplying both sides by 3, we increase the inequality consistently across both expressions, giving us \( 8 \times 3 > q \). It's important to note that multiplying or dividing both sides of an inequality by a positive number does not change the direction of the inequality sign. Simplifying results in \( 24 > q \), or more naturally expressed as \( q < 24 \). This step is fundamental since it reveals that \( q \) can be any number less than 24.

Once you've isolated the variable and rewritten the inequality, it's essential to verify the accuracy of your solution. This step ensures that the steps taken are correct and that no mistakes were made during calculations.

You do this by selecting a value that satisfies the resulting inequality. In this scenario, since \( q < 24 \), a straightforward choice could be \( q = 23 \). We substitute \( q = 23 \) back into the original inequality:

\( 8 > \frac{23}{3} \).

Calculating \( \frac{23}{3} \) gives us approximately 7.67. Comparing these values proves \( 8 > 7.67 \) is true. Since this inequality holds, \( q = 23 \) is indeed a solution, which validates our previous steps and confirms that our understanding and manipulation of the inequality were correct.

Simplifying expressions by eliminating fractions can make working with inequalities much more manageable. Fractions often complicate mathematical expressions and make them harder to interpret quickly.

To eliminate fractions, as in the exercise \( 8 > \frac{q}{3} \), multiply both sides by the denominator of the fraction. This step should be done carefully, paying attention to the sign of the number we multiply by. In this case, multiplying by 3 is safe because it is positive, and we maintain the inequality direction.

• This operation clears the fraction, leaving us with \( 8 \times 3 > q \).
• Simplifying results, we find \( 24 > q \).

By removing the fraction early on, the inequality becomes simpler, making it easier to isolate variables and understand potential solutions.