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The real number line is a simple straight line that represents all possible real numbers. Imagine it stretching infinitely in both directions, with zero in the middle. To the left, you'll find negative numbers, decreasing as you move further left. To the right are positive numbers, increasing as you go. Each point on this line corresponds to a unique real number.

In solving polynomial inequalities, the real number line helps us visualize where the solutions lie. By marking specific points, known as critical points, we can split the line into intervals and determine where the inequality holds true. These critical points divide the number line into different segments, each of which can be further tested for the inequality.

Critical points are specific values of a variable, often found by setting the polynomial expression to zero. For the inequality \((x-7)(x+3) \leq 0\), setting it equal to zero gives us the critical points \(x = 7\) and \(x = -3\). These points are key because they help divide the real number line into intervals that can be tested for the inequality.

When solving a polynomial inequality, knowing the critical points allows you to determine where the expression changes sign. Each critical point marks a potential switch from positive to negative (or vice versa). To solve, you must test each interval between these points to see where the inequality holds.

Interval notation is a concise way to express sets of solutions on the real number line. Instead of listing every number, we use brackets to indicate the start and end points of an interval.

• Use \([a, b]\) for closed intervals, where both endpoints are included.
• Use \((a, b)\) for open intervals, where endpoints are excluded.
• Mixed brackets, like \([a, b)\), indicate one endpoint is included and the other isn't.
• \((-∞, b)\) or \((a, ∞)\) show intervals extending infinitely.

In our inequality example, after testing the intervals formed by critical points \(-3\) and \(7\), we can express the solution set in interval notation by using brackets to show which parts of the line satisfy the inequality.

The solution set is the collection of all values that satisfy the given inequality. Once we determine which intervals are valid by testing points within each segment, we gather these into a complete set.

For a solution set of the inequality \((x-7)(x+3) \leq 0\), start by testing points in each interval formed by the critical points. For example, test points from \((-∞, -3)\), \((-3, 7)\), and \((7, ∞)\).

Only the intervals where the test points satisfy the inequality are included in the solution set. Combine these validated intervals using union notation, and express them in interval notation to complete the solution. This method ensures you capture all possible solutions on the real number line.