4.3 Lecture Guide: Solving Compound Inequalities
Objective: Identify an inequality that is a contradiction or an unconditional inequality.
The algebraic process for solving the inequalities we have examined in the first two sections of this chapter has left a variable term on one side of the inequality. These have all been conditional inequalities. Sometimes the algebraic process for solving an inequality will result in the variable being completely removed from the inequality, which means the inequality is a contradiction or an unconditional inequality.A _____________________ _____________________ is an inequality that is only true for certain values of the variable.
An _____________________ _____________________ is an inequality that is true for all values of the variable.
A __________________ is an inequality that is not true for any value of the variable.
5. 5 2 7 2 4x x x
Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1.
Solution: ____________
Type: __________________
10, 10, 1 by 10, 10, 1 1y
2y
6.
Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1.
Solution: ____________
Type: __________________
10, 10, 1 by 10, 10, 1
1y
2y
2 4 5 6 1x x x x
7.
Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1.
Solution: ____________
Type: __________________
10, 10, 1 by 10, 10, 1
1y
2y
2 3 5 9 6x x x
8.
Use the table and graph to determine the solution of each inequality. Then identify each inequality as a conditional inequality, a contradiction or an unconditional inequality. See Calculator Perspective 4.3.1.
Solution: ____________
Type: __________________
10, 10, 1 by 10, 10, 1
1y
2y
3 6 5 3 15 15x x x
Intersection of Two Sets
Algebraic Notation
A B
Verbally
The intersection of A and B is the set that contains the elements in both A and B.
Numerical Example
3,4 0,6 0,4
Graphical Example
Algebraic Notation
Verbally
Numerical Example
Graphical Example
Union of Two Sets
A B
The union of A and B is the set that contains the elements in either A or B or both.
3,4 0,6 3,6
9. Complete the following table.
Compound Inequality
Verbal Description
Graph Interval Notation
and
and
or
or
3x 7x
2x 0x
1 5x
4 7x
2x 5x
1x 2x
10. (a) Using the word ____________ between two inequalities indicates the intersection of two sets. In some cases, an intersection can be written in a combined form that looks like one expression sandwiched between two other expressions.
(b) Using the word ____________ between two inequalities indicates the union of two sets.
Graph each pair of intervals on the same number line and then give both their intersection
11. (3,6];A [ 1,4]B
A B = ____________
A B = ____________
A B and their union A B .
12.
A B = ____________
A B = ____________
( 3,9);A [2,17)B
Graph each pair of intervals on the same number line and then give both their intersection A B and their union A B .
13.
A B = ____________
A B = ____________
( ,6];A [1, )B
Graph each pair of intervals on the same number line and then give both their intersection A B and their union A B .
14.
A B = ____________
A B = ____________
( ,2);A (5, )B
Graph each pair of intervals on the same number line and then give both their intersection A B and their union A B .
15.
Write each inequality as two separate inequalities using the word “and” to connect the inequalities.
0 2x
16.
Write each inequality as two separate inequalities using the word “and” to connect the inequalities.
13 3x
Solve each compound inequality. Give the solution in interval notation.
26. 4 2 7 16x x 5 1 3 9x x and
27. Use the graph below to determine the solution of 2
6 1 53 3 3x x
x
-8
8
-8 8
y
x
2
21
3y x
1 63x
y
3 53x
y
Solution: ___________________