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Critical numbers play a key role in solving inequalities. These are the values of the variable, often represented as \(-4\) and \(3\) in our example, that mark the boundaries where the inequality might change its behavior. By analyzing the critical numbers, we get to know where the expression switches from being true to false or vice versa. In the inequality \(\frac{x + 4}{x - 3} \leq 0\), these numbers help us identify parts of the number line where the inequality holds.
To find critical numbers, set each part of the inequality to zero. For instance, in our case, setting \(x + 4 = 0\) gives \(-4\), and \(x - 3 = 0\) gives \(3\). These values help us verify the intervals that satisfy the inequality. Critical numbers are crucial because they act as checkpoints dividing the entire number line into distinct sections. Each section can then be evaluated to determine if it fits the inequality conditions.

Interval notation is a shorthand used to describe sets of numbers, or intervals on the number line. It's a neat way to show where the inequality is true. Different symbols in interval notation indicate whether endpoints are included or not.
In our example, \(( -\infty, -4 ) \cup [3, \infty)\) tells us:

• \((-\infty, -4)\) means all numbers less than \(-4\), and \(-4\) is not included.
• \([3, \infty)\) implies all numbers greater than or equal to \(3\), with \(3\) included.

The parentheses \(()\) indicate that the endpoint is not included, i.e., it is an open interval. Brackets \([]\) signify that the endpoint is included, making it a closed interval. As such, interval notation is a valuable tool in expressing ranges of solutions quickly and accurately, helping to visually identify solution sets for inequalities.

Solving inequalities, such as rational inequalities, involves just a few straightforward steps. The process involves examining the expression to identify where it is greater than or equal to zero, represented as \(\frac{x + 4}{x - 3} \leq 0\) in our task.
Here are the simple steps to follow:

• Find the critical numbers: Set the numerator and denominator to zero to identify these points.
• Sign test: Select test points from each interval created by the critical numbers to determine where the inequality holds true.
• Interval notation: Use results from the sign test to express which intervals satisfy the inequality.

Solving inequalities doesn't require guessing; it's about systematic testing and validation. These logical steps not only solve the inequality but also offer you insights into how the inequality behaves across the number line.