II. Graph and construct a sign chart for the following rational inequalities. \( \frac{x^{2}+4 x+3}{x-1}>0 \) Solution:

To graph and construct a sign chart for the inequality \( \frac{x^{2}+4 x+3}{x-1}>0 \), first, factor the numerator: \( x^{2}+4 x+3 = (x+1)(x+3) \). The critical points to consider are derived from both the numerator and denominator: \( x = -3, -1, 1 \). These points divide the number line into intervals: \( (-\infty, -3) \), \( (-3, -1) \), \( (-1, 1) \), and \( (1, +\infty) \). Testing values from each interval will tell you where the expression is positive or negative. Assuming you test \( x = -4, -2, 0, \) and \( 2 \): - For \( x = -4 \): positive. - For \( x = -2 \): negative. - For \( x = 0 \): positive. - For \( x = 2 \): positive. The sign chart shows which intervals satisfy the inequality, so we conclude that \( \frac{x^{2}+4 x+3}{x-1}>0 \) is true for the intervals \( (-\infty, -3) \), \( (-1, 1) \), and \( (1, +\infty) \). Let's not forget that \( x = 1 \) is an excluded point, as the inequality does not hold where the expression is undefined!