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Inequalities are like mathematical comparisons. They show how one value relates to another. Instead of using an equal sign, we use symbols like ">", "<", ">=", and "<=". These symbols tell us whether a number is smaller, larger, or possibly equal to another number.

In the exercise, we have the inequality \(x < 0\). This means we're interested in all numbers that are less than zero. Remember:

• "<" means less than
• "<=" means less than or equal to
• ">" means greater than
• ">=" means greater than or equal to

Recognizing these symbols helps you understand the relationship between numbers, allowing you to solve inequality problems with confidence.

The number line is a simple way to visualize numbers and their inequalities. It's a straight line where each point represents a real number. This visualization is helpful for understanding our inequality, \(x < 0\).

When using a number line:

• Mark the point 0 as a reference.
• Since \(x < 0\), we need to focus on the left side of 0.
• An empty circle is used at 0 because 0 is not part of our solution. This circle shows exclusion.

Finally, we shade everything to the left of 0. This shaded portion represents all the numbers that satisfy the inequality. The number line effectively shows which numbers are included or excluded from the solution set.

Graphing inequalities is a way to visually represent solutions on the number line. For our inequality \(x < 0\), we used an open circle at 0 to show that 0 is not included. Next, we shaded the line to the left, representing all numbers smaller than 0. This graphical representation makes it easy to see solutions at a glance.

Now, let's introduce interval notation. Interval notation provides a concise way to write the set of solutions. For \(x < 0\), the interval notation is \((-fty, 0)\). Both \(-fty\) and 0 are not included, which is why we use parentheses.

In interval notation:

• Parentheses \(()\) mean the number is not included.
• Brackets \([]\) mean the number is included.
• \(-\infty\) and \(\infty\) are always written with parentheses, as they are not actual numbers.

Understanding how to graph inequalities and express them using interval notation helps in interpreting and solving various mathematical problems.