Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5. (x + 1)/(x - 5) > 0

Verified step by step guidance

1. Step 1: Identify the critical points of the inequality. These are the values of x that make the numerator or the denominator equal to zero. In this case, the critical points are x = -1 (from the numerator) and x = 5 (from the denominator).
2. Step 2: Use these critical points to divide the number line into intervals. The intervals are (-∞, -1), (-1, 5), and (5, ∞).
3. Step 3: Test a number from each interval in the inequality to determine if the inequality is true or false. For example, you could test x = -2, x = 0, and x = 6.
4. Step 4: If the inequality is true for a test value in an interval, then all values in that interval are solutions to the inequality. If the inequality is false for a test value in an interval, then no values in that interval are solutions.
5. Step 5: Write the solution set in interval notation. Include the intervals where the inequality is true, and exclude the intervals where the inequality is false. Remember that you cannot include x = 5 in the solution set because it makes the denominator zero, which is undefined.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to an inequality (e.g., >, <). To solve them, one must determine where the rational expression is positive or negative, which often requires finding critical points where the numerator or denominator equals zero.

Critical Points

Critical points are values of the variable that make the numerator or denominator of a rational expression zero. These points divide the number line into intervals, which can be tested to determine where the inequality holds true. For the inequality (x + 1)/(x - 5) > 0, the critical points are x = -1 and x = 5.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the solution set for the inequality can be expressed in interval notation to clearly show the values of x that satisfy the inequality.