Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. (x+1)/(x-4)>0

Verified step by step guidance

1. <Step 1: Identify the critical points by setting the numerator and denominator equal to zero.>
2. <Step 2: Solve the equation x + 1 = 0 to find the critical point from the numerator.>
3. <Step 3: Solve the equation x - 4 = 0 to find the critical point from the denominator.>
4. <Step 4: Use the critical points to divide the number line into intervals.>
5. <Step 5: Test each interval to determine where the inequality (x+1)/(x-4)>0 holds true, and express the solution in interval notation.>

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to zero, typically using inequality symbols like >, <, ≥, or ≤. To solve these inequalities, one must determine where the rational expression is positive or negative, which often requires finding critical points where the expression equals zero or is undefined.

Critical Points

Critical points are values of the variable that make the rational expression equal to zero or undefined. For the inequality (x+1)/(x-4)>0, the critical points are found by setting the numerator (x+1) to zero and the denominator (x-4) to zero. These points divide the number line into intervals that can be tested to determine where the inequality holds true.

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.