absolute inequalities-algebraic

1. More on Absolute Value Inequalities
(Algebraic Method–Optional)

2. More on Absolute Value Inequalities
Since |x| means “the distance between x and 0”, so the
expression |x| < c means “the distance between x and 0 is
less than c”.

3. Example A. Draw the inequality |x| < 7.
Since |x| means “the distance between x and 0”, so the
expression |x| < c means “the distance between x and 0 is
less than c”.
More on Absolute Value Inequalities

4. Example A. Draw the inequality |x| < 7.
Since |x| means “the distance between x and 0”, so the
expression |x| < c means “the distance between x and 0 is
less than c”.
We are to draw all numbers which are within 7 units from the
number 0.
More on Absolute Value Inequalities

5. Example A. Draw the inequality |x| < 7.
-7-7 7
0
x
Since |x| means “the distance between x and 0”, so the
expression |x| < c means “the distance between x and 0 is
less than c”.
We are to draw all numbers which are within 7 units from the
number 0.
x
More on Absolute Value Inequalities

6. Example A. Draw the inequality |x| < 7.
-7 < x < 7-7-7 7
0
x
Since |x| means “the distance between x and 0”, so the
expression |x| < c means “the distance between x and 0 is
less than c”.
We are to draw all numbers which are within 7 units from the
number 0.
x
More on Absolute Value Inequalities

7. Example A. Draw the inequality |x| < 7.
-7 < x < 7-7-7 7
0
x
Since |x| means “the distance between x and 0”, so the
expression |x| < c means “the distance between x and 0 is
less than c”.
We are to draw all numbers which are within 7 units from the
number 0.
x
The open circles means the end points are not included in
the solution.
More on Absolute Value Inequalities

8. I. (One piece | |–inequalities)
Example A. Draw the inequality |x| < 7.
-7 < x < 7-7-7 7
0
x
Since |x| means “the distance between x and 0”, so the
expression |x| < c means “the distance between x and 0 is
less than c”.
We are to draw all numbers which are within 7 units from the
number 0.
x
The open circles means the end points are not included in
the solution.
More on Absolute Value Inequalities

9. I. (One piece | |–inequalities)
If |x| < c then –c < x < c.
Example A. Draw the inequality |x| < 7.
-7 < x < 7-7-7 7
0
x
Since |x| means “the distance between x and 0”, so the
expression |x| < c means “the distance between x and 0 is
less than c”.
We are to draw all numbers which are within 7 units from the
number 0.
x
The open circles means the end points are not included in
the solution.
More on Absolute Value Inequalities

10. I. (One piece | |–inequalities)
If |x| < c then –c < x < c.
Example A. Draw the inequality |x| < 7.
-7 < x < 7-7-7 7
0
x
Since |x| means “the distance between x and 0”, so the
expression |x| < c means “the distance between x and 0 is
less than c”.
We are to draw all numbers which are within 7 units from the
number 0.
x
In general, if we have
|expression| < c
The open circles means the end points are not included in
the solution.
More on Absolute Value Inequalities

11. I. (One piece | |–inequalities)
If |x| < c then –c < x < c.
Example A. Draw the inequality |x| < 7.
-7 < x < 7-7-7 7
0
x
Since |x| means “the distance between x and 0”, so the
expression |x| < c means “the distance between x and 0 is
less than c”.
We are to draw all numbers which are within 7 units from the
number 0.
x
In general, if we have
|expression| < c
we rewrite it without the "| |" as
– c < expression < c.
The open circles means the end points are not included in
the solution.
More on Absolute Value Inequalities

12. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
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13. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
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14. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
b. | x2 – 2x + 1| < 1
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15. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
b. | x2 – 2x + 1| < 1
–1 <x2 – 2x + 1 < 1
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16. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
b. | x2 – 2x + 1| < 1
–1 <x2 – 2x + 1 < 1
Example C. Solve the inequality |3 – 2x| < 7 and draw the
solution.
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17. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
b. | x2 – 2x + 1| < 1
–1 <x2 – 2x + 1 < 1
Example C. Solve the inequality |3 – 2x| < 7 and draw the
solution.
Rewrite the inequality without the | | as
–7 < 3 – 2x < 7
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18. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
b. | x2 – 2x + 1| < 1
–1 <x2 – 2x + 1 < 1
Example C. Solve the inequality |3 – 2x| < 7 and draw the
solution.
Rewrite the inequality without the | | as
–7 < 3 – 2x < 7 subtract 3 from each part
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19. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
b. | x2 – 2x + 1| < 1
–1 <x2 – 2x + 1 < 1
Example C. Solve the inequality |3 – 2x| < 7 and draw the
solution.
Rewrite the inequality without the | | as
–7 < 3 – 2x < 7 subtract 3 from each part
–10 < –2x < 4
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b. | x2 – 2x + 1| < 1

20. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
b. | x2 – 2x + 1| < 1
–1 <x2 – 2x + 1 < 1
Example C. Solve the inequality |3 – 2x| < 7 and draw the
solution.
Rewrite the inequality without the | | as
–7 < 3 – 2x < 7 subtract 3 from each part
–10 < –2x < 4 divide by -2, need to switch
the inequality around
More on Absolute Value Inequalities

21. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
b. | x2 – 2x + 1| < 1
–1 <x2 – 2x + 1 < 1
Example C. Solve the inequality |3 – 2x| < 7 and draw the
solution.
Rewrite the inequality without the | | as
–7 < 3 – 2x < 7 subtract 3 from each part
–10 < –2x < 4 divide by -2, need to switch
the inequality around
–10/–2 > –2x/–2 > 4/–2
More on Absolute Value Inequalities

22. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
b. | x2 – 2x + 1| < 1
–1 <x2 – 2x + 1 < 1
Example C. Solve the inequality |3 – 2x| < 7 and draw the
solution.
Rewrite the inequality without the | | as
–7 < 3 – 2x < 7 subtract 3 from each part
–10 < –2x < 4 divide by -2, need to switch
the inequality around
–10/–2 > –2x/–2 > 4/–2
5 > x > –2
More on Absolute Value Inequalities

23. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
b. | x2 – 2x + 1| < 1
–1 <x2 – 2x + 1 < 1
Example C. Solve the inequality |3 – 2x| < 7 and draw the
solution.
Rewrite the inequality without the | | as
–7 < 3 – 2x < 7 subtract 3 from each part
–10 < –2x < 4 divide by -2, need to switch
the inequality around
–10/–2 > –2x/–2 > 4/–2
5 > x > –2
-2 5
0
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24. The expression |x| > c means “the distance from x to 0 is
more than c”.
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25. Example D. Draw the inequality |x| > 7.
The expression |x| > c means “the distance from x to 0 is
more than c”.
More on Absolute Value Inequalities

26. Example D. Draw the inequality |x| > 7.
The expression |x| > c means “the distance from x to 0 is
more than c”.
We are to draw all x’s which are 7 or more units from the
number 0.
More on Absolute Value Inequalities

27. Example D. Draw the inequality |x| > 7.
The expression |x| > c means “the distance from x to 0 is
more than c”.
We are to draw all x’s which are 7 or more units from the
number 0.
x < –7 or 7 < x
-7 70
More on Absolute Value Inequalities

28. Example D. Draw the inequality |x| > 7.
The expression |x| > c means “the distance from x to 0 is
more than c”.
We are to draw all x’s which are 7 or more units from the
number 0. This includes the end points 7 and –7.
x < –7 or 7 < x
-7 70
More on Absolute Value Inequalities

29. Example D. Draw the inequality |x| > 7.
The expression |x| > c means “the distance from x to 0 is
more than c”.
We are to draw all x’s which are 7 or more units from the
number 0. This includes the end points 7 and –7.
x < –7 or 7 < x
-7 70
The solid circles means the end points are part of the solution.
More on Absolute Value Inequalities

30. II. (Two–piece | |–inequalities)
Example D. Draw the inequality |x| > 7.
The expression |x| > c means “the distance from x to 0 is
more than c”.
We are to draw all x’s which are 7 or more units from the
number 0. This includes the end points 7 and –7.
x < –7 or 7 < x
-7 70
The solid circles means the end points are part of the solution.
More on Absolute Value Inequalities

31. Example D. Draw the inequality |x| > 7.
More on Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is
more than c”.
We are to draw all x’s which are 7 or more units from the
number 0. This includes the end points 7 and –7.
x < –7 or 7 < x
-7 70
The solid circles means the end points are part of the solution.
II. (Two–piece | |–inequalities)
If |x| > c then x < –c or that c < x.
More on Absolute Value Inequalities

32. Example D. Draw the inequality |x| > 7.
The expression |x| > c means “the distance from x to 0 is
more than c”.
We are to draw all x’s which are 7 or more units from the
number 0. This includes the end points 7 and –7.
x < –7 or 7 < x
-7 70
-c c0
c < xx< –c
The solid circles means the end points are part of the solution.
II. (Two–piece | |–inequalities)
If |x| > c then x < –c or that c < x.
More on Absolute Value Inequalities

33. Example D. Draw the inequality |x| > 7.
The expression |x| > c means “the distance from x to 0 is
more than c”.
We are to draw all x’s which are 7 or more units from the
number 0. This includes the end points 7 and –7.
In general, if we have the inequality
|expression| > c
we drop the | | and rewrite it as two inequalities
expression < – c or c < expression
x < –7 or 7 < x
-7 70
-c c0
c < xx< –c
The solid circles means the end points are part of the solution.
II. (Two–piece | |–inequalities)
If |x| > c then x < –c or that c < x.
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34. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
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35. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
Drop the | |, write it as two inequalities as
x + y < –2 or 2 < x + y
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36. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
Drop the | |, write it as two inequalities as
x + y < –2 or 2 < x + y
B. | x2 – 2x + 1| > 1
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37. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
Drop the | |, write it as two inequalities as
x + y < –2 or 2 < x + y
B. | x2 – 2x + 1| > 1
Drop the | |, write it as two inequalities as
x2 – 2x + 1 < –1 or 1 < x2 – 2x + 1
More on Absolute Value Inequalities

38. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
Drop the | |, write it as two inequalities as
x + y < –2 or 2 < x + y
B. | x2 – 2x + 1| > 1
Drop the | |, write it as two inequalities as
x2 – 2x + 1 < –1 or 1 < x2 – 2x + 1
Example F. Solve the inequality |3x – 2| > 4. Draw.
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39. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
Drop the | |, write it as two inequalities as
x + y < –2 or 2 < x + y
B. | x2 – 2x + 1| > 1
Drop the | |, write it as two inequalities as
x2 – 2x + 1 < –1 or 1 < x2 – 2x + 1
Example F. Solve the inequality |3x – 2| > 4. Draw.
Rewrite the inequality as two inequalities without the | | as
3x – 2 < - 4 or 4 < 3x – 2
More on Absolute Value Inequalities

40. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
Drop the | |, write it as two inequalities as
x + y < –2 or 2 < x + y
B. | x2 – 2x + 1| > 1
Drop the | |, write it as two inequalities as
x2 – 2x + 1 < –1 or 1 < x2 – 2x + 1
Example F. Solve the inequality |3x – 2| > 4. Draw.
Rewrite the inequality as two inequalities without the | | as
3x – 2 < - 4 or 4 < 3x – 2
3x < -2
More on Absolute Value Inequalities

41. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
Drop the | |, write it as two inequalities as
x + y < –2 or 2 < x + y
B. | x2 – 2x + 1| > 1
Drop the | |, write it as two inequalities as
x2 – 2x + 1 < –1 or 1 < x2 – 2x + 1
Example F. Solve the inequality |3x – 2| > 4. Draw.
Rewrite the inequality as two inequalities without the | | as
3x – 2 < - 4 or 4 < 3x – 2
3x < -2
x < -2/3
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42. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
Drop the | |, write it as two inequalities as
x + y < –2 or 2 < x + y
B. | x2 – 2x + 1| > 1
Drop the | |, write it as two inequalities as
x2 – 2x + 1 < –1 or 1 < x2 – 2x + 1
Example F. Solve the inequality |3x – 2| > 4. Draw.
Rewrite the inequality as two inequalities without the | | as
3x – 2 < - 4 or 4 < 3x – 2
3x < -2 6 < 3x
x < -2/3
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43. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
Drop the | |, write it as two inequalities as
x + y < –2 or 2 < x + y
B. | x2 – 2x + 1| > 1
Drop the | |, write it as two inequalities as
x2 – 2x + 1 < –1 or 1 < x2 – 2x + 1
Example F. Solve the inequality |3x – 2| > 4. Draw.
Rewrite the inequality as two inequalities without the | | as
3x – 2 < - 4 or 4 < 3x – 2
3x < -2 6 < 3x
x < -2/3 or 2 < x
More on Absolute Value Inequalities

44. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
Drop the | |, write it as two inequalities as
x + y < –2 or 2 < x + y
B. | x2 – 2x + 1| > 1
Drop the | |, write it as two inequalities as
x2 – 2x + 1 < –1 or 1 < x2 – 2x + 1
Example F. Solve the inequality |3x – 2| > 4. Draw.
Rewrite the inequality as two inequalities without the | | as
3x – 2 < - 4 or 4 < 3x – 2
3x < -2 6 < 3x
x < -2/3 or 2 < x
-2/3 20
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45. Ex. Translate the expressions algebraically and solve.
Draw the solution.
1. |x| < 2 2. |x| < 5 3. |–x| < 2 4. |–x| ≤ 5
5. |x| ≥ –2 6. |–2x| < 6 7. |–3x| ≥ 6 8. |–x| ≥ –5
9. |3 – x| ≥ –5 10. |3 + x| ≤ 7 11. |x – 9| < 5
12. |5 – x| < 5 13. |4 + x| ≥ 9 14. |x + 1| ≥ 3
21. |4 – 3x| ≤ 3 22. |3 + 2x| < 7 23. |–2x + 3| > 5
24. |4 – 2x| ≤ –3 25. |2x + 7| < 5 26. 3|2x + 1| ≤ 5
15. |x – 2| < 1 16. |3 – x| ≤ 5 17. |x – 5| < 5
18. |7 – x| < 3 19. |1 – 2x| < 9 20. |2x + 1| < 3
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