Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+4)(x−1)/(x+2)≤0

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1. Identify the critical points by setting the numerator and denominator equal to zero separately. For the numerator \((x+4)(x-1)\), set \(x+4=0\) and \(x-1=0\) to find \(x=-4\) and \(x=1\). For the denominator \(x+2\), set \(x+2=0\) to find \(x=-2\).
2. Plot the critical points \(-4\), \(-2\), and \(1\) on a number line. These points divide the number line into four intervals: \((-\infty, -4)\), \((-4, -2)\), \((-2, 1)\), and \((1, \infty)\).
3. Determine the sign of the rational expression \(\frac{(x+4)(x-1)}{x+2}\) on each interval by choosing a test point from each interval and substituting it into the expression. Remember that the expression is undefined at \(x=-2\) because the denominator is zero there.
4. Since the inequality is \(\leq 0\), include intervals where the expression is negative or zero. Also, include points where the numerator is zero (i.e., \(x=-4\) and \(x=1\)) because the expression equals zero there, but exclude \(x=-2\) where the expression is undefined.
5. Write the solution set in interval notation based on the intervals where the inequality holds true, and graph these intervals on the real number line, using closed dots for included endpoints and open dots for excluded points.

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 where the denominator is not 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.

Interval Notation and Graphing on the Number Line

Interval notation concisely represents solution sets using parentheses and brackets to indicate open or closed intervals. Graphing on the number line visually shows these intervals, marking included points with solid dots and excluded points with open circles, aiding in understanding the solution.