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Interval notation is a method used to describe the solution set of inequalities on a number line. It’s a shorthand way of capturing all numbers that make an inequality true. If you're new to it, here's how it works:

• Use parentheses \( ( \) or \( ) \) for numbers that are not included in the range, like an open circle on a graph.
• Use brackets \[ [ \] or \[ ] \] for numbers that are included, similar to a closed circle on a graph.

Take the inequality \(x^2 + 6x \geq -9\). Once simplified and solved, the solution is all real numbers because \( (x + 3)^2 \geq 0 \) for all \(x\). This is represented in interval notation as \((-\infty, \infty)\). This notation implies that every number from negative infinity to positive infinity satisfies the inequality. A useful tip: whenever you see \((-\infty, \infty)\), it means there are no restrictions on \(x\).

Factoring quadratics is a crucial skill in algebra. It involves rewriting a quadratic expression as a product of its binomial factors. This technique can simplify solving equations and inequalities.

Let's break down the process:

• A quadratic expression takes the form \(ax^2 + bx + c\).
• To factor it, find two numbers that multiply to \(a c\) and add up to \(b\).
• Rewrite the middle term using these numbers, then factor by grouping.

For the expression \(x^2 + 6x + 9\), you notice it’s a perfect square trinomial. You rewrite it as \((x+3)^2\). Here, \((x+3)\) is the repeated factor. Perfect squares are excellent because they simplify to a form where you can square root if needed, making them easier to handle in inequalities. This set up is perfect for solving the inequality as shown in the problem, allowing us to conclude that \((x+3)^2 \geq 0\) confirms non-negativity across all real numbers.

Graphing inequalities on a number line visually demonstrates the range of solutions. For the inequality \(x^2 + 6x \geq -9\), we found the solution to be all real numbers \((-\infty, \infty)\). Here's how you would graph that:

• Identify the range: For \((-\infty, \infty)\), the solution includes every possible point on the number line.
• Shade the line: Whenever the solution is \((-\infty, \infty)\), you shade the entire line. This tells the viewer that every real number satisfies the inequality.
• Use open or closed circles: If your interval notation for the graph required it, display open circles to exclude boundary points or closed circles to include them.

Graphing brings clarity to a solution set, giving a visual confirmation of the algebraic solution. It’s especially helpful when your range is less straightforward than here, such as when solving inequalities with finite solutions or when using combinations of intervals.