Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. $(x^2-x-2)/(x^2-4x+3)>0$

Verified step by step guidance

1. First, factor both the numerator and the denominator of the rational expression. For the numerator \(x^2 - x - 2\), find two numbers that multiply to \(-2\) and add to \(-1\). For the denominator \(x^2 - 4x + 3\), find two numbers that multiply to \$3\( and add to \)-4$.
2. Rewrite the inequality using the factored forms: \(\frac{(x - a)(x - b)}{(x - c)(x - d)} > 0\), where \(a\), \(b\), \(c\), and \(d\) are the roots found from factoring.
3. Identify the critical points by setting each factor equal to zero: \(x = a\), \(x = b\), \(x = c\), and \(x = d\). These points divide the real number line into intervals.
4. Determine the sign of the expression on each interval by choosing a test point from each interval and substituting it into the factored inequality. Remember that the expression is undefined at points where the denominator is zero, so exclude those from the solution set.
5. Based on the sign analysis, write the solution set where the expression is greater than zero. Then, graph this solution set on the real number line, marking excluded points (where the denominator is zero) with open circles.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the numerator and denominator change sign, considering domain restrictions where the denominator is zero.

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals zero, dividing the number line into intervals. By testing points in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Domain Restrictions and Excluded Values

Since division by zero is undefined, values that make the denominator zero must be excluded from the solution set. Identifying these points ensures the solution respects the domain of the rational expression and avoids invalid solutions.