Download 1.2 One-Variable, Compound, and Absolute Value Inequalities

1.2 One-Variable, Compound,
and Absolute Value Inequalities
Interval Notation
Interval notation is another way to represent the
solution set. Interval notation uses open
parentheses ( ) and closed parentheses [ ]. Use
the open parentheses if the value is not
included in the graph and closed parentheses if
the value is included.
Conjunction
A conjunction is a compound statement formed
by joining two statements with the connector
“and.” A conjunction is true when both of its
combined parts are true; otherwise it is false.
Disjunction
A disjunction is a compound statement formed
by joining two statements with the connector
“or.” A disjunction is false if and only if both
statements are false; otherwise it is true.
Union of Sets
The union of sets A and B, written A B, is the
set of elements that are members of set A or
of set B or of both sets. This is used to denote
a disjunction.
Example 1
Graph the set.
1. [–2, 5)
2.
Multiplying and Dividing by A
Negative with Inequalities
If you multiply or divide both sides of an
inequality by a negative quantity, the inequality
symbol is reversed.
Example 2
Solve and graph. Write answers in interval
notation.
Example 2
Solve and graph. Write answers in interval
notation.
Example 2
Solve and graph. Write answers in interval
notation.
Example 2
Solve and graph. Write answers in interval
notation.
Solving an Absolute Value Inequality
If X is an algebraic expression and c is a positive
number,
1. The solutions of
are the numbers that
satisfy
.
2. The solutions of
are the numbers that
satisfy
Steps for Solving an Absolute Value
Inequality
1. Completely isolate the absolute value.
2. To create the first inequality, remove the
absolute value symbol.
3. To create the second inequality, flip the
inequality symbol and change the value of
the constant.
4. Solve both inequalities.
5. Graph and write the solution in interval
notation.
Example 3
Solve each absolute value inequality. Write
answers in interval notation.
Example 3
Solve each absolute value inequality. Write
answers in interval notation.
Example 3
Solve each absolute value inequality. Write
answers in interval notation.
Example 3
Solve each absolute value inequality. Write
answers in interval notation.
Example 3
Solve each absolute value inequality. Write
answers in interval notation.
1.3 Complex Numbers
Imaginary Unit i
The imaginary unit i is defined as
Complex Numbers and Imaginary
Numbers
Set of all numbers in the form
a + bi
where a, b are real numbers and i is the
imaginary unit.
Note: If b is radical write i first.
Standard Form
Adding and Subtracting Complex Numbers
1. (a + bi) + (c + di) = (a + c) + (b + d)i
2. (a + bi) – (c + di) = (a – c) + (b – d)i
Example 1
Perform the indicated operations, writing the
results in standard form:
a. (5 – 11i) + (7 + 4i)
Example 1
Perform the indicated operations, writing the
results in standard form:
b. (–5 + i) – (–11 – 6i)
Multiplying Complex Numbers
When you multiply complex numbers, use the
distributive property and/or the FOIL method.
After completing the multiplication, we
replace any occurrences of
Example 2
Find the products:
a. 4i(3 – 5i)
b. (7 – 3i)( –2 – 5i)
Conjugate of a Complex Number
The multiplication of complex conjugates gives
a real number.
Example 3
Divide and express the results in standard
form.
Example 3
Divide and express the results in standard
form.
Principal Square Root of a Negative
Number
For any positive real number b, the principle
square root of the negative number –b is
defined by
**When performing operations with square
roots of negative numbers, first express all
square roots in terms of i.
Example 4
Perform the indicated operations and write
the result in standard form:
Example 4
Perform the indicated operations and write
the result in standard form:
Example 4
Perform the indicated operations and write
the result in standard form:
Example 4
Perform the indicated operations and write
the result in standard form:
Homework #2
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107-111 odd
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