Calculus- Early Transcendentals, 2021a

10 Review
Inequality Rules
Add/subtract a number to both sides:
•Ifa < b,thena + c < b + c and a − c < b − c.
Adding two inequalities of the same type:
•Ifa < b and c < d, thena + c < b + d.
Add the left sides together, add the right sides together.
Multiplying by a positive number:
•Letc > 0. If a < b,thenc · a < c · b.
Multiplying by a negative number:
•Letc < 0. If a < b,thenc · a > c · b.
Note that we reversed the inequality symbol!
Similar rules hold for each of ≤, > and ≥.
Solving Basic Inequalities
We can use the inequality rules to solve some simple inequalities.
Example 1.10: Basic Inequality
Find all values of x satisfying
3x + 1 > 2x − 3
Write your answer in both interval and set-builder notation. Finally, draw a number line indicating
your solution set.
Solution. Subtracting 2x from both sides gives x + 1 > −3. Subtracting 1 from both sides gives x > −4.
Therefore, the solution is the interval (−4,∞). In set-builder notation the solution may be written as
{x ∈ R : x > −4}. We illustrate the solution on the number line as follows:
Sometimes we need to split our inequality into two cases as the next example demonstrates.
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Example 1.11: Double Inequalities
Solve the inequality
4 > 3x − 2 ≥ 2x − 1