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Understanding critical values is key when solving rational inequalities. These values are crucial because they indicate where the expression is either zero or undefined. For the inequality \(\frac{3x - 5}{x - 5} \geq 0\), the critical values are determined by examining both the numerator and the denominator separately.

• Zero of the numerator: The numerator \(3x - 5\) becomes zero when \(3x - 5 = 0\). Solving this equation gives the critical value \(x = \frac{5}{3}\). At this point, the entire fraction becomes zero, which meets the \(\geq 0\) condition.
• Undefined point of the expression: The denominator \(x - 5\) becomes zero at \(x = 5\), which makes the entire expression undefined. Dividing by zero is not allowed in mathematics, hence the inequality is not defined here.

These critical values help in dividing the number line into intervals for further testing to find the solution set.

Graphing inequalities helps visualize the solution space effectively by using the critical values identified earlier. For \(\frac{3x - 5}{x - 5} \geq 0\), we first list the intervals derived from the critical values. These are

• \((-\infty, \frac{5}{3})\)
• \((\frac{5}{3}, 5)\)
• \((5, \infty)\)

To graph the inequality, you test each interval by picking test points:
- For \((-\infty, \frac{5}{3})\), choose \(x = 0\); thus, the fraction becomes \(\frac{-5}{-5} = 1\), which is greater than or equal to zero.- For \((\frac{5}{3}, 5)\), with \(x = 2\), you get \(\frac{1}{-3}\), which is not greater than zero.- For \((5, \infty)\), picking \(x = 6\) gives \(\frac{13}{1}\), which indeed is greater than zero.
Graphically, this means \((-\infty, \frac{5}{3}]\) and \((5, \infty)\) are part of the solution set, indicated by filling or shading these intervals on a number line. Remember to use an open circle for \(x = 5\) to indicate this value is not included in the solution set, whereas a closed circle is used at \(x = \frac{5}{3}\) to include this critical value.

The solution set for a rational inequality indicates where the inequality is satisfied, and it's crucial in interpreting results. In the context of our example \(\frac{3x - 5}{x - 5} \geq 0\), the solution set is expressed in interval notation and tells us the values of \(x\) that make the inequality true.The process involves:

• Inclusion of Critical Values: Since the inequality is \(\geq 0\), \(x = \frac{5}{3}\) is included in the solution set as it makes the fraction exactly zero, satisfying the \(\geq\) part of the inequality.
• Exclusion of Undefined Points: \(x = 5\) is excluded because it makes the denominator zero, leading to an undefined expression.

Thus, the final solution set \((-\infty, \frac{5}{3}] \cup (5, \infty)\) reflects all \(x\) values that do not violate these conditions, with the union \(\cup\) indicating that the solution consists of all parts that meet the inequality across different intervals. This complete interval notation helps in succinctly conveying the range of values that solve the inequality and can be effectively illustrated on the number line.