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Interval notation is a way of expressing a set of numbers along a number line. It is particularly helpful because it conveys which numbers are included and excluded in a concise manner.
When you see an inequality like \(x < -18\), the solution set includes all numbers less than \(-18\). In interval notation, this is written as \((-fty, -18)\).
Here, the negative infinity symbol \(-\infty\) indicates that the solution extends indefinitely to the left on the number line. Parentheses \(()\) are used rather than brackets \([]\) because \(-\infty\) and \(-18\) are not part of the solution. This notation tells us that the values are approaching but never actually reaching \(-\infty\) or \(-18\).

• The left part of the interval, \(-\infty\), implies going endlessly far into negative numbers.
• The right part of the interval, \(-18\), signals that the endpoint is not included in the solution set since there is an open circle at \(-18\) on the graph.

When dealing with fractions in equations or inequalities, it's often useful to clear them out by finding a least common denominator (LCD). The LCD is the smallest number that both denominators can divide into evenly, and it helps to simplify expressions.
In the inequality \(\frac{1}{3}x + 2 < \frac{1}{6}x - 1\), the denominators are 3 and 6, making the LCD 6. By multiplying each term in the inequality by this common denominator, we remove the fractions completely:

• Multiply the term \(\frac{1}{3}x\) by 6 to get \(2x\).
• Multiply the term \(2\) by 6 to get \(12\).
• Multiply the term \(\frac{1}{6}x\) by 6 to get \(x\).
• Multiply the term \(-1\) by 6 to get \(-6\).

This yields the simplified inequality of \(2x + 12 < x - 6\). Eliminating fractions simplifies the problem and allows us to focus on the inequality itself without the complexities of fractional arithmetic.

Graphing inequalities is a visual way of representing solution sets on a number line. It shows which numbers make the inequality true.
For \(x < -18\), we start with a number line. Here’s how to graph it:

• Identify the critical number, which in this case, is \(-18\).
• Place an open circle on \(-18\) on the number line to show that \(-18\) itself is not included in the solution. An open circle means the endpoint is excluded.
• Shade the line to the left of \(-18\) to indicate that all numbers smaller than \(-18\) satisfy the inequality.

The shaded portion represents the infinite set of solutions that satisfy \(x < -18\). This approach gives a quick, intuitive view of the range of possible solutions, helping to solidify the understanding of inequalities.