Calculus- Early Transcendentals, 2021a

8 Review
1.1.4 Inequalities, Intervals and Solving Basic Inequalities
Inequality Notation
Recall that we use the symbols ,≤,≥ when writing an inequality. In particular,
• a < b means a is to the left of b (that is, a is strictly less than b),
• a ≤ b means a is to the left of or the same as b (that is, a is less than or equal to b),
• a > b means a is to the right of b (that is, a is strictly greater than b),
• a ≥ b means a is to the right of or the same as b (that is, a is greater than or equal to b).
To keep track of the difference between the symbols, some students use the following mnemonic.
Mnemonic
The < symbol looks like a slanted L which stands for “Less than”.
Example 1.9: Inequalities
The following expressions are true:
1 < 2, 5 < −2, 1 ≤ 2, 1 ≤ 1, 4 ≥ π > 3, 7.23 ≥−7.23.
The real numbers are ordered and are often illustrated using the real number line:
Intervals
Assume a,b are real numbers with a < b (i.e., a is strictly less than b). An interval is a set of every real
number between two indicated numbers and may or may not contain the two numbers themselves. When
describing intervals we use both round brackets and square brackets.
(1) Use of round brackets in intervals: ( , ). The notation (a,b) is what we call the open interval from
atoband consists of all the numbers between a and b, but does not include a or b. Using set-builder
notation we write this as:
(a,b)={x ∈ R : a < x < b}
We read {x ∈ R : a < x < b} as “the set of real numbers x such that x is greater than a and less than b”On
the real number line we represent this with the following diagram: