Alg1, Unit 11, Lesson01_absent-student, page 1
Review of one-dimensional inequalities
Compound inequalities
Recall that inequalities are solved with the same techniques with which
equations are solved… with one exception: if both sides are multiplied
or divided by a negative number, the inequality symbol must be
reversed.
Example 1: Solve 4(–2x – 3) < 20. Give the answer both algebraically and
graphically on a number line.
Compound inequalities are two or more inequalities joined by either
the words “and” or “or.”
x ≥ -6 and x < 0
(also written as x ≥ -6 ∩ x < 0)
conjunction
x < 11 or x > 22 (also written as x < 11 ∪ x > 22) disjunction
The word “or” must always be explicitly written; however, “and” is
often implied:
3 < x ≤ 5, implies
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Alg1, Unit 11, Lesson01_absent-student, page 2
“Or” means to take the union of the two inequalities.
A simple view of union (symbolized with ∪) is, “take it all.”
Example 2: Graph the solution to x < -6 or x > 11.
Example 3: Graph the solution to x > 2 ∪ x ≥ 12.
Example 4: Express this graph as a compound inequality.
x < -2 or x ≥ 6
“And” means to take the intersection of the two inequalities.
A simple view of intersection (symbolized with ∩) is, “where the
two graphs overlap (what they have in common).”
Example 5: Graph the solution to the compound inequality given by 8 ≤ x < 15.
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Alg1, Unit 11, Lesson01_absent-student, page 3
Example 6: Solve the inequality 3 < x – 4 < 8. Give the answer both algebraically
and graphically.
Example 7: Express this graph as a compound inequality.
-2 ≤ x < 6
Example 8: Express the answer to Example 7 using either ∪ or ∩.
(-2 ≤ x) ∩ (x< 6)
Union and intersection can also be applied to sets. For example, if
set1= {a, b, c, d} and set2 = {c, d, e, f}, then
set1 ∪ set2 = {a, b, c, d, e, f}
set1 ∩ set2 = {c, d}
A strange syntax:
Sometimes compound inequalities are written like this:
{ x| 4 < x < 20 }
This is read, “x, such that 4 is less than x which is less than 20”,
and simply means, “x is between 4 and 20 (exclusive on both
ends).”
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Alg1, Unit 11, Lesson01_absent-student, page 4
Assignment:
1. Write the compound inequality represented by this graph.
2. Write the compound inequality represented by this graph.
3. Represent 5 < x ≤ 15 as a graph on a
number line.
4. Represent -3 < x and x < 10 as a
graph on a number line.
5. Represent x < 4 or x > 8 as a graph
on a number line.
6. Represent x > -5 ∪ x > 3 as a graph
on a number line.
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Alg1, Unit 11, Lesson01_absent-student, page 5
7. If the score is predicted to be
somewhere between 20 and 42, write
this score as a compound inequality.
8. Graph the compound inequality in
problem 7 on a number line.
9. If the weight of an object is said to
greater than 56 lbs and at the same
time less than or possibly equal to 100
lbs, express the weight as a compound
inequality.
10. Graph the compound inequality in
problem 9 on a number line.
11. If the slope of a line is either less
than -2 or greater than 5, express the
slope as a compound inequality.
12. Graph the compound inequality in
problem 11 on a number line.
13. If the cost is at least $50, express
the cost as an inequality.
14. If the cost is at most $150, express
the cost as an inequality.
15. Graph the solution to this inequality on a number line: 14 < x ≤ 22
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Alg1, Unit 11, Lesson01_absent-student, page 6
16. Graph the solution to this compound inequality on a number line:
-100 < x ∩ x < 200
17. Graph the solution to this compound inequality on a number line:
x < 0 or x < 9
18. Graph the solution to this compound inequality on a number line:
x > 11 or x < 0
19. Graph the solution to 3x – 1 > 2 on a number line.
20. Graph the solution to 4 – 6x ≥ -30 on a number line.
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Alg1, Unit 11, Lesson01_absent-student, page 7
In problems 21-24, state whether the inequality statements are true or false.
21. 5 < 22
22. -6 > -5
*23. x > 2x where x is known to be a
positive number.
*24. x > 2x where x is known to be a
negative number.
*25. Find the solution to this compound inequality and display the answer on a
number line: 30 < (2 – x)6 and x + 1 ≤ 0
26. If s1 = {apple, orange, mango, pear} and s2 = {turnip, apple, radish, plum}, find
s1 ∩ s2 and s1 ∪ s2.
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