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Interval notation is a way of describing a set of numbers on the real number line. It is often used to express the solution sets of inequalities in a compact and clear format. This method uses brackets (like \([ \ ]\) and \(( \ )\)) to show whether endpoints are included or excluded in the set.

- A square bracket \([\) or \(]\) indicates that an endpoint is included, also called *inclusive*. For example, \([-2, 3]\) includes all numbers from -2 to 3, including -2 and 3.
- A parenthesis \((\) or \()\) indicates that an endpoint is not included, which is called *exclusive*. For instance, \((-2, 3)\) includes all numbers between -2 and 3, but does not include -2 and 3 themselves.
- The infinity symbol \((\infty\) or \( -\infty)\) is always paired with a parenthesis because infinity is not a number you can "reach" or include. It represents the idea that the number line continues indefinitely.

When expressing solutions of inequalities like \(x \leq -\frac{10}{9}\), interval notation allows us to write the solution as \(( -\infty, -\frac{10}{9}]\) easily. It conveys that all numbers from \(-\infty\) up to and including \(-\frac{10}{9}\) are part of the solution.

Graphing inequalities involves visually representing the solution set of an inequality on a number line. It helps to clearly show which numbers satisfy the inequality conditions.

Here's how you graph inequalities like \(x \leq -\frac{10}{9}\):

• First, draw a horizontal line to represent the number line.
• Locate and mark the point \(-\frac{10}{9}\) on the line. Since our inequality includes \(-\frac{10}{9}\) (indicated by \(\leq\)), we put a solid dot at this point.
• For inequalities with a greater than or equal to sign, we shade to the right to include all larger numbers; however, since we have a less than or equal to inequality \(x \leq -\frac{10}{9}\), we shade to the left. This tells us all values of \(x\) that are less than or equal to \(-\frac{10}{9}\) form part of the solution set.

Graphing an inequality provides a handy visual reference, helping one quickly understand which values make the inequality true.

A number line is a graphical representation of numbers along a straight line. It's a fundamental concept in mathematics, making it easy to visualize basic arithmetic operations and inequalities.

Here's why a number line is useful in graphing inequalities:

• It provides a visual way to see the range of possible solutions.
• You can easily show inclusivity and exclusivity of endpoints using dots: solid for included and hollow for not included.
• For complex solutions involving several inequalities, the number line helps in seeing overlapping solution sets at a glance.

For the inequality \(x \leq -\frac{10}{9}\), the number line helps us plot \(-\frac{10}{9}\) correctly and shows us shading in the correct region (to the left). By utilizing a number line for graphing inequalities, students can see the solution's span intuitively, bridging the gap between symbolic representation and visualization.