Calculus- Early Transcendentals, 2021a

12 Review
Guidelines for Solving Rational Inequalities
1. Move everything to one side to get a 0 on the other side.
2. If needed, combine terms using a common denominator.
3. Factor the numerator and denominator.
4. Identify points where either the numerator or denominator is 0. Such points are called split
points.
5. Draw a number line and indicate your split points on the number line. Draw closed/open
circles for each split point depending on if that split point satisfies the inequality (division by
zero is not allowed).
6. The split points will split the number line into subintervals. For each subinterval pick a test
point and see if the expression in Step 3 is positive or negative. Indicate this with a + or −
symbol on the number line for that subinterval.
7. Now write your answer in set-builder notation. Use the union symbol ∪ if you have multiple
intervals in your solution.
Example 1.13: Rational Inequality
Write the solution to the following inequality using interval notation:
2 − x
2 + x ≥ 1.
Solution. One method to solve this inequality is to multiply both sides by 2 + x, but because we do not
know if 2+x is positive or negative we must split it into two cases (Case 1: 2+x > 0 and Case 2: 2+x < 0).
Instead we follow the guidelines for solving rational inequalities:
Start with original problem:
Move everything to one side:
Find a common denominator:
Combine fractions:
Expand numerator:
Simplify numerator:
2 − x
2 + x ≥ 1
2 − x
2 + x − 1 ≥ 0
2 − x
2 + x − 2 + x
2 + x ≥ 0
(2 − x) − (2 + x)
2 + x
2 − x − 2 − x
2 + x
≥ 0
−2x
2 + x ≥ 0 (∗)
≥ 0