# Domain of Function by mikesanye

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```									      Section 4.3

Solving Compound Inequalities
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.
Solve each inequality. Identify each contradiction or
unconditional inequality.
1. 3x  2  x  4  2 x
Solve each inequality. Identify each contradiction or
unconditional inequality.
2. 5x  3  2 x  3x  3
Solve each inequality. Identify each contradiction or
unconditional inequality.
3. 4  2 x  3  7 x  12  x
Solve each inequality. Identify each contradiction or
unconditional inequality.
4. 3 x  4  2 x  5  x  1
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.

5.   5x  2  7 x  2  4  x 

Solution: ____________

Type: __________________

y2
y1
10, 10, 1 by 10, 10, 1
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.

6. 2 x   x  4   5 x  6  x  1

Solution: ____________

Type: __________________

y2

y1

10, 10, 1 by 10, 10, 1
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.

7. 2  x  3  5 x  9  6 x

Solution: ____________

Type: __________________
y1

y2

10, 10, 1 by 10, 10, 1
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.

8. 3 6 x  5   3x  15  15 x

Solution: ____________

Type: __________________
y1

y2
10, 10, 1 by 10, 10, 1
Objective: Solve compound inequalities involving
intersection and union.
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
Union of Two Sets

Algebraic Notation
A B

Verbally
The union of A and B is the set that contains the elements in
either A or B or both.

Numerical Example

 3, 4   0,6   3,6
Graphical Example
9. Complete the following table.

Compound           Verbal        Graph   Interval
Inequality         Description           Notation
x  3 and x  7

x  2 and x  0

1  x  5

4 x7
x  2 or x  5
x  1 or x  2
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 A  B and their union A  B.

11. A  (3,6]; B  [1, 4]
A  B = ____________

A  B = ____________
Graph each pair of intervals on the same number line and
then give both their intersection A  B and their union A  B.

12. A  (3,9); B  [2,17)
A  B = ____________

A  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  (,6]; B  [1,  )
A  B = ____________

A  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  (, 2); B  (5, )
A  B = ____________

A  B = ____________
Write each inequality as two separate inequalities using the
word “and” to connect the inequalities.

15.   0 x2
Write each inequality as two separate inequalities using the
word “and” to connect the inequalities.

16.   13  x  3
Write each inequality expression as a single compound
inequality.

17. x  2 and x  3
Write each inequality expression as a single compound
inequality.

18. x  10 and x  8
Solve each compound inequality. Give the solution in interval
notation.
19. 12  6 x  24
Solve each compound inequality. Give the solution in interval
notation.
20.   1  2 x  1  3
Solve each compound inequality. Give the solution in interval
notation.
x
21. 1   3  4
2
Solve each compound inequality. Give the solution in interval
notation.
22. 5x  2  6 x  1  5x  4
Solve each compound inequality. Give the solution in interval
notation.
23. 3x  2  x  1 or 4 x  1  3  x  2 
Solve each compound inequality. Give the solution in interval
notation.

24. 3x  2  6  x and 3x  2  12  x
Solve each compound inequality. Give the solution in interval
notation.
x x
25. 2 x  3  5 x  9 or   1
2 3
Solve each compound inequality. Give the solution in interval
notation.
26. 5x  1  3x  9 and 4 x  2  7 x  16
27. Use the graph below to determine the solution of
x     2        x
  6  x 1    5
3     3        3
y
2
y2  x  1
8

3
x
y1    6
3
x
-8                     8

x
-8                   y3    5
3

Solution: ___________________
28. The perimeter of the parallelogram shown must be at
least 20 cm and no more than 48 cm. If the given length
must be 5 cm, determine the possible lengths for x, the
unknown dimension.

x cm

5 cm                             5 cm

x cm

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