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Critical points in rational inequalities are where the rational function equals zero. A rational function is described by a numerator and a denominator.

• The numerator creates a critical point when it equals zero because this is where the entire fraction has the potential to equal zero.
• The denominator cannot be zero, but it indicates a vertical asymptote or an undefined point in the inequality setup.

To find the critical points, set both the numerator and the denominator equal to zero and solve for the variable in each case. For the inequality \( \frac{3x+5}{6-2x} \geq 0 \), solve the equation \( 3x + 5 = 0 \), to find one critical point, \( x = -\frac{5}{3} \). For the denominator, \( 6 - 2x = 0 \) gives another critical point at \( x = 3 \). These points help to determine the intervals on a number line where the inequality will be evaluated.

Interval notation is a concise way of writing a range of numbers, often used to describe solution sets precisely and efficiently. It comprises endpoints and sometimes parentheses or brackets to indicate whether endpoints are included in the interval.

• Brackets, \([ \text{ and } ]\), mean the endpoint is included in the interval.
• Parentheses, \(( \text{ and } )\), indicate the endpoint is not included.

In the case of the rational inequality, The solution set obtained was \(-\frac{5}{3} \leq x < 3\). This uses a bracket on \(-\frac{5}{3}\) because 'equal to' is part of the inequality specification, and a parenthesis on 3 since it cannot make the denominator zero. Thus, the solution does not include 3 itself but includes all values approaching it.

A solution set of an inequality includes all of the values that satisfy the inequality condition. To find a solution set for a rational inequality:1. Identify critical points by setting the numerator and denominator equal to zero.2. Use these points to divide the number line into regions or intervals.3. Test a point from each interval to see if it satisfies the inequality.The solution set for \( \frac{3x+5}{6-2x} \geq 0 \) is \(-\frac{5}{3} \leq x < 3\). This is the interval where any point you test within will maintain the inequality's truth. Thus, the solution set includes all numbers between \(-\frac{5}{3}\) and \(3\), excluding \(3\) itself, captured elegantly in the interval notation.

The real number line is a visual representation of all possible solutions that can exist for a given inequality. It stretches from negative infinity to positive infinity, encompassing all integers, fractions, and irrational numbers in between.

• Placing critical points on the real number line helps to visualize the problem. They break the number line into segments or intervals that you need to test.
• A solid dot on the line indicates inclusion of a number, while an open dot shows exclusion.

In solving \( \frac{3x+5}{6-2x} \geq 0 \), the real number line reveals two crucial areas: one between \(-\frac{5}{3}\) and \(3\) where the inequality is true, and numbers outside these points where it is false. Seeing the solution set on this line solidifies understanding by clearly showing which sections satisfy the inequality and contribute to the final solution.