Math 152 — Rodriguez
Blitzer — 4.2
Compound Inequalities
I. Compound Inequalities in one variable
A compound inequality consists of two inequalities joined by the word ‘and’ or ‘or’.
II. Intersection and Union of Sets
A. The intersection of sets A and B consists of all numbers that are in BOTH set A AND
in set B. It is denoted by A ∩ B.
Example:
A= {1, 2, 3, 4}
A∩B
B = {2, 4, 6, 8}
= {2, 4} because 2 and 4 are in BOTH set A AND in set B
Examples: A= {1, 2, 3, 4}
A= {1, 2, 3, 4}
B = {2, 3, 4, 5}
A∩B=
B = {8}
A∩B=
B. The union of sets A and B consists of all numbers that are in set A OR in set B OR in
both sets. It is denoted by A ∪ B.
Example:
A= {1, 2, 3, 4}
B = {2, 4, 6, 8}
A ∪ B = {1, 2, 3, 4, 6, 8}
because these numbers are in set A OR set B OR in both sets
Examples: A= {1, 2, 3, 4}
A= {1, 2, 3}
B = {2, 3, 4, 5}
B = {11, 12}
A∪B=
A∪B=
C. The main idea is that you want to associate "AND" with the intersection of two sets
and "OR" with the union of two sets.
and → intersection → numbers that are in BOTH sets
or → union → numbers that are in one set or the other set or in both sets; that
is numbers that are in ANY of the sets
Compound
Inequality
x ≥ 8 and x≥ 3
x ≥ 8 or x ≥ 3
Scratch work
Graph of solution set
Interval
Notation
D. Examples:
Compound
Inequality
Scratch work
Graph of solution set
Interval
Notation
x < 5 and x ≥ 1
x < 5 or x ≥ 1
x < 2 and x > 4
x < 2 or x > 4
III. Solving Compound Inequalities with "and" or “or”
Steps to solve compound inequalities:
1. Solve each linear inequality separately.
2. Graph each inequality on a number line.
3. a) If the conjunction is “and”: look for the intersection of these two sets; numbers
that are in BOTH sets
b) If the conjunction is “or”: look for the union of these two sets; numbers that are
in ANY set (one set or the other set or both sets)
4. Make a separate graph that shows only the solutions to the compound inequality.
5. Write the solution set in interval notation.
Examples: Solve. Graph the solution set. Other than the empty set, express the solution
set in interval notation.
1) x ! 1 " 8 and 4x + 1 ! 9
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2)
x ! 1 " 8 or 4x + 1 ! 9
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3)
2x ! 1 " 9 or 3x ! 2 > 4
4)
x ! 3 < 1 and 2x ! 3 " 9
IV. Solving Compound Inequalities of the form a < bx + c < d
A. The compound inequality
a < bx + c < d
a < bx + c
and
is “shorthand” for
bx + c < d
B. To solve this type of inequality you have TWO choices:
Choice A) Rewrite them as two separate inequalities (so we follow the steps in part III)
Choice B) Solve them “as is”: goal is to isolate the variable in the “middle” part of the
compound inequality. To do this, whatever we do to the “middle” part we do
to the two “outer” parts of the compound inequality.
Example: Solve:
Solve “as is”:
3 < 2x + 4 < 8
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3 < 2x + 4 < 8
Solve by rewriting as two separate inequalities:
3 < 2x + 4 and 2x+4 < 8
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V. Now you try: Solve. Graph the solution set. Other than the empty set, express the
solution set in interval notation.
1) 2x + 1 > 4x ! 3 and x ! 1 " 3x + 5
2) 2x + 1 > 4x ! 3 or x ! 1 " 3x + 5
3) 2x ! 5 " !11 or 5x + 1 ! 6
4) 2x ! 5 " !11 and 5x + 1 ! 6
5) !8 < 2x ! 5 " 11
6) Let f(x)= x—2 and g(x) = 2x+1.
Find all values of x for which
f(x)≤5 or g(x)>3.
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