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Interval notation is a way to represent a set of numbers, especially useful when dealing with inequalities. In interval notation, parentheses \(( )\) and brackets \([ ]\) are used to indicate whether the endpoints are included or excluded from the set.
For instance, in the inequality \(x < \frac{1}{2}\), all values for \(x\) must be less than \(\frac{1}{2}\), but not equal to \(\frac{1}{2}\). Therefore, in interval notation, the solution is written as \((-\infty, \frac{1}{2})\). The open parenthesis \((\) indicates that \(\frac{1}{2}\) is not included in the solution set.

• "-\infty" means that the set continues indefinitely in the negative direction.
• Use parentheses for values not included in the set, such as in \((-\infty, \frac{1}{2})\).

This concise representation helps communicate the solution effectively without the need for lengthy descriptions.

Graphing inequalities is a visual way to illustrate the solution set on a number line. This process makes it easier to understand which values of a variable satisfy the inequality.
Consider the inequality \(x < \frac{1}{2}\). To graph this:

• Draw a horizontal number line, marking the significant point, \(\frac{1}{2}\).
• Place an open circle on \(\frac{1}{2}\), because \(\frac{1}{2}\) itself is not included in the solution set.
• Shade or draw an arrow extending to the left from this open circle, towards negative infinity, indicating all values less than \(\frac{1}{2}\) are part of the solution.

These visual clues help to better comprehend which numbers are valid solutions and which are not, especially at a glance.

Fraction inequalities, such as \(\frac{4}{5} x < \frac{2}{5}\), often seem complex but can be solved effectively with a clear approach.

• **Step 1: Eliminate the Fraction**
  Multiply every term by the denominator. For example, multiply both sides by 5 to clear the fractions: \[5 \cdot \frac{4}{5} x < 5 \cdot \frac{2}{5}\] This simplifies to \(4x < 2\).
• **Step 2: Solve the Simplified Inequality**
  Divide each side by 4 to solve for \(x\): \[x < \frac{1}{2}\]

The process of eliminating fractions and solving the resulting simpler inequality ensures clarity and precision when finding solutions to these types of inequalities.