Solve each rational inequality. Give the solution set in interval notation. $\frac{x^2-3x-4}{x^2+6x+9}\le 0$

Verified step by step guidance

1. First, identify the rational inequality: \(\frac{x^2 - 3x - 4}{x^2 + 6x + 9} \leq 0\).
2. Factor both the numerator and the denominator: numerator \(x^2 - 3x - 4\) factors as \((x - 4)(x + 1)\), and denominator \(x^2 + 6x + 9\) factors as \((x + 3)^2\).
3. Determine the critical points by setting numerator and denominator equal to zero: numerator zeros at \(x = 4\) and \(x = -1\), denominator zero at \(x = -3\) (note this makes the denominator zero, so \(x = -3\) is excluded from the domain).
4. Create a number line and mark the critical points \(-3\), \(-1\), and \(4\). Test values from each interval formed by these points in the inequality to determine where the expression is less than or equal to zero.
5. Write the solution set by including intervals where the inequality holds true, remembering to exclude points where the denominator is zero, and express the solution in interval notation.

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 expression is positive, negative, or zero, considering the domain restrictions from the denominator.

Factoring Polynomials

Factoring is the process of breaking down polynomials into products of simpler polynomials. It helps identify zeros of the numerator and denominator, which are critical points for determining intervals to test in the inequality.

Interval Testing and Domain Restrictions

After finding critical points, the number line is divided into intervals. Each interval is tested to see if the inequality holds. Additionally, values that make the denominator zero are excluded from the solution set because they are undefined.