The document discusses absolute value inequalities and provides examples of how to represent them graphically. It explains that an inequality of the form |x| < c can be rewritten as -c < x < c, representing all values within c units of 0. An inequality of the form |x| > c is split into two inequalities x < -c or c < x, representing values more than c units from 0. Examples are given of drawing the solutions to |x| < 7, |x| > 7, and solving |3 - 2x| < 7 algebraically then graphically.
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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
More on Absolute Value Inequalities
13. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
More on Absolute Value Inequalities
14. Example B. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
b. | x2 – 2x + 1| < 1
More on Absolute Value Inequalities
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
More on Absolute Value Inequalities
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.
More on Absolute Value Inequalities
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
More on Absolute Value Inequalities
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
More on Absolute Value Inequalities
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
More on Absolute Value Inequalities
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
More on Absolute Value Inequalities
24. The expression |x| > c means “the distance from x to 0 is
more than c”.
More on Absolute Value Inequalities
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.
More on Absolute Value Inequalities
34. Example E. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
More on Absolute Value Inequalities
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
More on Absolute Value Inequalities
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
More on Absolute Value Inequalities
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.
More on Absolute Value Inequalities
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
More on Absolute Value Inequalities
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
More on Absolute Value Inequalities
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
More on Absolute Value Inequalities