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An inequality is a mathematical statement that compares two expressions using inequality symbols like <, >, ≤, or ≥. In our problem, we have the inequality:

1. \(x < 30\)

This inequality states that \(x\) can be any value less than 30. In other words, \(x\) can be negative, zero, or any number smaller than 30, but it cannot be 30 itself.

A few important points to remember about inequalities:

• If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol.
• Inequalities describe a range of values, which can often be theorized using interval notation and visuals like number lines.

The solution set for an inequality includes all values that satisfy the inequality. Let's figure out the solution set for our problem:

The inequality: \(x < 30\) means all numbers less than 30 are part of the solution set. These numbers together form what we call the solution set.

To express this set in a neat and formal way, we use interval notation.

### Interval Notation
- Interval notation is a way of writing subsets of the real number line
- It uses brackets and parentheses to show where intervals start and end
For \(x < 30\):

• The interval starts at \(-\infty\) because there is no lower bound to \( x \)
• It ends at 30, but not including 30 itself, indicated by the parenthesis '(', so we use: \((-\infty, 30)\)

Therefore, the interval notation for the solution set of \(x<30\) is \((-\infty, 30)\)

Graphing inequalities on a number line provides a visual way to represent the solution set. For the given inequality \(x < 30\), let's go through the steps to graph it:

1. **Draw a number line:** Draw a horizontal line with numbers marked based on your inequality.

2. **Locate the critical value:** Find and mark the number 30 on the line.

3. **Open circle:** Place an open circle at 30. This shows that 30 itself is not included in the solution set. Open circles are used for `<` and `>` inequalities, while closed circles are used for `≤` and `≥` inequalities.

4. **Shading:** Shade the number line to the left of 30. This shading represents all the numbers less than 30.

Using these steps to graph \(x < 30\), the visual makes it clear that every number to the left of 30 and not including 30 itself is part of the solution set.

Graphing inequalities is a useful tool to help understand the range of possible values and provides a quick visual summary.