The document discusses absolute value and distance on the real number line. It provides examples to illustrate that the distance between two numbers x and y is defined as the absolute value of their difference, |x - y|. This ensures the distance is always positive or zero. When the values of x and y are unknown, both x - y and y - x could represent their distance, so absolute value is used to unambiguously refer to the positive value. Graphically, an absolute value inequality like |x - c| < r represents all values within a distance r of the center c, between c - r and c + r.

1. Absolute Value and Distance

2. Absolute Value and Distance
If we don’t know the values of x and y, then the distance
between them on the real line is either x – y or y – x.

3. Absolute Value and Distance
If we don’t know the values of x and y, then the distance
between them on the real line is either x – y or y – x.
For example, if x = 5 and y = 2, then the distance between
them is x – y = 3 where as y – x = –3 is negative.

4. Absolute Value and Distance
If we don’t know the values of x and y, then the distance
between them on the real line is either x – y or y – x.
For example, if x = 5 and y = 2, then the distance between
them is x – y = 3 where as y – x = –3 is negative.
To clarify the two choices for the distance, we put the absolute
value symbol “l l” around the expression as l x – y l,
to remind us “to take the positive (or 0) one” as
the distance between x and y.

5. Absolute Value and Distance
If we don’t know the values of x and y, then the distance
between them on the real line is either x – y or y – x.
For example, if x = 5 and y = 2, then the distance between
them is x – y = 3 where as y – x = –3 is negative.
To clarify the two choices for the distance, we put the absolute
value symbol “l l” around the expression as l x – y l,
to remind us “to take the positive (or 0) one” as
the distance between x and y. Facts about the “l l”:
1. l # l ≥ 0 or “distance” is non–negative.
2. l x – y l = l y – x l = the distance between x and y
so “distance” is symmetric (mutual).
3. With y = 0, l x l = l –x l = the distance from x (or –x) to 0.

6. Absolute Value and Distance
If we don’t know the values of x and y, then the distance
between them on the real line is either x – y or y – x.
For example, if x = 5 and y = 2, then the distance between
them is x – y = 3 where as y – x = –3 is negative.
To clarify the two choices for the distance, we put the absolute
value symbol “l l” around the expression as l x – y l,
to remind us “to take the positive (or 0) one” as
the distance between x and y. Facts about the “l l”:
1. l # l ≥ 0 or “distance” is non–negative.
2. l x – y l = l y – x l = the distance between x and y
so “distance” is symmetric (mutual).
3. With y = 0, l x l = l –x l = the distance from x (or –x) to 0.
So l x l = r (r > 0) has two answers, x =±r, as shown.
0 r–r

7. Absolute Value and Distance
If we don’t know the values of x and y, then the distance
between them on the real line is either x – y or y – x.
For example, if x = 5 and y = 2, then the distance between
them is x – y = 3 where as y – x = –3 is negative.
To clarify the two choices for the distance, we put the absolute
value symbol “l l” around the expression as l x – y l,
to remind us “to take the positive (or 0) one” as
the distance between x and y. Facts about the “l l”:
1. l # l ≥ 0 or “distance” is non–negative.
2. l x – y l = l y – x l = the distance between x and y
so “distance” is symmetric (mutual).
3. With y = 0, l x l = l –x l = the distance from x (or –x) to 0.
So l x l = r (r > 0) has two answers, x =±r, as shown.
Hence l x l ≠ r are the rest of the line,
which consists of three pieces, 0 r–r

8. Absolute Value and Distance
If we don’t know the values of x and y, then the distance
between them on the real line is either x – y or y – x.
For example, if x = 5 and y = 2, then the distance between
them is x – y = 3 where as y – x = –3 is negative.
To clarify the two choices for the distance, we put the absolute
value symbol “l l” around the expression as l x – y l,
to remind us “to take the positive (or 0) one” as
the distance between x and y. Facts about the “l l”:
1. l # l ≥ 0 or “distance” is non–negative.
2. l x – y l = l y – x l = the distance between x and y
so “distance” is symmetric (mutual).
3. With y = 0, l x l = l –x l = the distance from x (or –x) to 0.
So l x l = r (r > 0) has two answers, x =±r, as shown.
Hence l x l ≠ r are the rest of the line,
which consists of three pieces,
the "center“ interval
0 r–r

9. Absolute Value and Distance
If we don’t know the values of x and y, then the distance
between them on the real line is either x – y or y – x.
For example, if x = 5 and y = 2, then the distance between
them is x – y = 3 where as y – x = –3 is negative.
To clarify the two choices for the distance, we put the absolute
value symbol “l l” around the expression as l x – y l,
to remind us “to take the positive (or 0) one” as
the distance between x and y. Facts about the “l l”:
1. l # l ≥ 0 or “distance” is non–negative.
2. l x – y l = l y – x l = the distance between x and y
so “distance” is symmetric (mutual).
3. With y = 0, l x l = l –x l = the distance from x (or –x) to 0.
So l x l = r (r > 0) has two answers, x =±r, as shown.
Hence l x l ≠ r are the rest of the line,
which consists of three pieces,
the "center“ interval and the two flanks extend to ±∞.
0 r–r

10. Example A.
a. Translate the meaning of |x| < 7.
Draw the solutions.
Absolute Value Inequalities

11. Example A.
a. Translate the meaning of |x| < 7.
Draw the solutions.
|x| < 7
the distance between x and 0
Absolute Value Inequalities

12. Example A.
a. Translate the meaning of |x| < 7.
Draw the solutions.
|x| < 7
the distance between x and 0 is less than 7.
Absolute Value Inequalities

13. Example A.
a. Translate the meaning of |x| < 7.
Draw the solutions.
These are all the numbers which
are within 7 units from 0, from –7 to 7.
|x| < 7
the distance between x and 0 is less than 7.
Absolute Value Inequalities
–7 < x < 7-7-7 70

14. Example A.
a. Translate the meaning of |x| < 7.
Draw the solutions.
These are all the numbers which
are within 7 units from 0, from –7 to 7.
|x| < 7
the distance between x and 0 is less than 7.
b. Translate the meaning of |x| ≥ 7.
Draw the solutions.
Absolute Value Inequalities
–7 < x < 7-7-7 70

15. Example A.
a. Translate the meaning of |x| < 7.
Draw the solutions.
These are all the numbers which
are within 7 units from 0, from –7 to 7.
|x| < 7
the distance between x and 0 is less than 7.
b. Translate the meaning of |x| ≥ 7.
Draw the solutions.
|x| ≥ 7
the distance between x and 0 is at least 7.
Absolute Value Inequalities
–7 < x < 7-7-7 70

16. Example A.
a. Translate the meaning of |x| < 7.
Draw the solutions.
These are all the numbers which
are within 7 units from 0, from –7 to 7.
|x| < 7
the distance between x and 0 is less than 7.
b. Translate the meaning of |x| ≥ 7.
Draw the solutions.
These are all the numbers which are 7 units or more from 0
and they are the two shoulders on the line as shown.
|x| ≥ 7
the distance between x and 0 is at least 7.
Absolute Value Inequalities
–7 < x < 7-7-7 70

17. Example A.
a. Translate the meaning of |x| < 7.
Draw the solutions.
These are all the numbers which
are within 7 units from 0, from –7 to 7.
|x| < 7
the distance between x and 0 is less than 7.
b. Translate the meaning of |x| ≥ 7.
Draw the solutions.
x < –7 –7 7
0
These are all the numbers which are 7 units or more from 0
and they are the two shoulders on the line as shown.
|x| ≥ 7
the distance between x and 0 is at least 7.
7 < x
Absolute Value Inequalities
–7 < x < 7-7-7 70

18. I. (One Piece | |–Inequalities)
If |x| < r then –r < x < r.
Absolute Value Inequalities

19. I. (One Piece | |–Inequalities)
If |x| < r then –r < x < r.
Absolute Value Inequalities
Recall that |x – c| translates into “the distance between x and c”
so the expression |x – c| < r means the
“the distance between x and c is less than r”.

20. I. (One Piece | |–Inequalities)
If |x| < r then –r < x < r.
Absolute Value Inequalities
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
Recall that |x – c| translates into “the distance between x and c”
so the expression |x – c| < r means the
“the distance between x and c is less than r”.

21. I. (One Piece | |–Inequalities)
If |x| < r then –r < x < r.
Absolute Value Inequalities
|x – 2| < 3
the distance between x and 2 is less than 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
Recall that |x – c| translates into “the distance between x and c”
so the expression |x – c| < r means the
“the distance between x and c is less than r”.

22. I. (One Piece | |–Inequalities)
If |x| < r then –r < x < r.
Absolute Value Inequalities
|x – 2| < 3
the distance between x and 2 is less than 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
Recall that |x – c| translates into “the distance between x and c”
so the expression |x – c| < r means the
“the distance between x and c is less than r”.
2

23. I. (One Piece | |–Inequalities)
If |x| < r then –r < x < r.
Absolute Value Inequalities
|x – 2| < 3
the distance between x and 2 is less than 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
Recall that |x – c| translates into “the distance between x and c”
so the expression |x – c| < r means the
“the distance between x and c is less than r”.
2–3= –1 2+3=5
2
right 3left 3

24. I. (One Piece | |–Inequalities)
If |x| < r then –r < x < r.
Absolute Value Inequalities
|x – 2| < 3
the distance between x and 2 is less than 3
or –1 < x < 5.
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
Recall that |x – c| translates into “the distance between x and c”
so the expression |x – c| < r means the
“the distance between x and c is less than r”.
2–3= –1 2+3=5
2
right 3left 3

25. I. (One Piece | |–Inequalities)
If |x| < r then –r < x < r.
Absolute Value Inequalities
|x – 2| < 3
the distance between x and 2 is less than 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
Recall that |x – c| translates into “the distance between x and c”
so the expression |x – c| < r means the
“the distance between x and c is less than r”.
In general |x – c| < r is the
1–dimensional "circle“ on the line,
centered at c with radius r,
extending from c – r to c + r.
2–3= –1 2+3=5
2
right 3left 3or –1 < x < 5.

26. I. (One Piece | |–Inequalities)
If |x| < r then –r < x < r.
Absolute Value Inequalities
|x – 2| < 3
the distance between x and 2 is less than 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
Recall that |x – c| translates into “the distance between x and c”
so the expression |x – c| < r means the
“the distance between x and c is less than r”.
In general |x – c| < r is the
1–dimensional "circle“ on the line,
centered at c with radius r,
extending from c – r to c + r.
c
2–3= –1 2+3=5
2
right 3left 3or –1 < x < 5.

27. I. (One Piece | |–Inequalities)
If |x| < r then –r < x < r.
Absolute Value Inequalities
|x – 2| < 3
the distance between x and 2 is less than 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
Recall that |x – c| translates into “the distance between x and c”
so the expression |x – c| < r means the
“the distance between x and c is less than r”.
In general |x – c| < r is the
1–dimensional "circle“ on the line,
centered at c with radius r,
extending from c – r to c + r.
c
r r
2–3= –1 2+3=5
2
right 3left 3or –1 < x < 5.

28. I. (One Piece | |–Inequalities)
If |x| < r then –r < x < r.
Absolute Value Inequalities
|x – 2| < 3
the distance between x and 2 is less than 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| < 3 and solve.
Recall that |x – c| translates into “the distance between x and c”
so the expression |x – c| < r means the
“the distance between x and c is less than r”.
In general |x – c| < r is the
1–dimensional "circle“ on the line,
centered at c with radius r,
extending from c – r to c + r.
c
r r
2–3= –1 2+3=5
2
right 3left 3or –1 < x < 5.
c+rc–r

29. I. (Two–Piece | |–Inequalities)
If |x| > r then the solution
are {x < –r} U {x < r}.
Absolute Value Inequalities
0 r–r
|x| > r:

30. I. (Two–Piece | |–Inequalities)
If |x| > r then the solution
are {x < –r} U {x < r}.
Absolute Value Inequalities
The expression |x – c| > r means the “the distance from x to c is
more than r”.
0 r–r
|x| > r:

31. I. (Two–Piece | |–Inequalities)
If |x| > r then the solution
are {x < –r} U {x < r}.
Absolute Value Inequalities
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| ≥ 3 and solve.
The expression |x – c| > r means the “the distance from x to c is
more than r”. They are the two flanks outside of the circle.
0 r–r
|x| > r:

32. I. (Two–Piece | |–Inequalities)
If |x| > r then the solution
are {x < –r} U {x < r}.
Absolute Value Inequalities
|x – 2| ≥ 3
the distance between x and 2 is at least 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| ≥ 3 and solve.
The expression |x – c| > r means the “the distance from x to c is
more than r”. They are the two flanks outside of the circle.
0 r–r
|x| > r:

33. I. (Two–Piece | |–Inequalities)
If |x| > r then the solution
are {x < –r} U {x < r}.
Absolute Value Inequalities
|x – 2| ≥ 3
the distance between x and 2 is at least 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| ≥ 3 and solve.
The expression |x – c| > r means the “the distance from x to c is
more than r”. They are the two flanks outside of the circle.
2–3= –1 2+3=5
2
right 3left 3
0 r–r
|x| > r:

34. I. (Two–Piece | |–Inequalities)
If |x| > r then the solution
are {x < –r} U {x < r}.
Absolute Value Inequalities
|x – 2| ≥ 3
the distance between x and 2 is at least 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| ≥ 3 and solve.
The expression |x – c| > r means the “the distance from x to c is
more than r”. They are the two flanks outside of the circle.
2–3= –1 2+3=5
2
right 3left 3or {x ≤ –1 } U {5 ≤ x}.
0 r–r
|x| > r:

35. I. (Two–Piece | |–Inequalities)
If |x| > r then the solution
are {x < –r} U {x < r}.
Absolute Value Inequalities
|x – 2| ≥ 3
the distance between x and 2 is at least 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| ≥ 3 and solve.
The expression |x – c| > r means the “the distance from x to c is
more than r”. They are the two flanks outside of the circle.
In general |x – c| ≥ r consists of the numbers outside of the
"circle” centered at c with radius r,
i.e. {x ≤ c – r} U {c + r ≤ x}.
2–3= –1 2+3=5
2
right 3left 3or {x ≤ –1 } U {5 ≤ x}.
0 r–r
|x| > r:

36. I. (Two–Piece | |–Inequalities)
If |x| > r then the solution
are {x < –r} U {x < r}.
Absolute Value Inequalities
|x – 2| ≥ 3
the distance between x and 2 is at least 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| ≥ 3 and solve.
The expression |x – c| > r means the “the distance from x to c is
more than r”. They are the two flanks outside of the circle.
In general |x – c| ≥ r consists of the numbers outside of the
"circle” centered at c with radius r,
i.e. {x ≤ c – r} U {c + r ≤ x}. c
2–3= –1 2+3=5
2
right 3left 3or {x ≤ –1 } U {5 ≤ x}.
0 r–r
|x| > r:

37. I. (Two–Piece | |–Inequalities)
If |x| > r then the solution
are {x < –r} U {x < r}.
Absolute Value Inequalities
|x – 2| ≥ 3
the distance between x and 2 is at least 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| ≥ 3 and solve.
The expression |x – c| > r means the “the distance from x to c is
more than r”. They are the two flanks outside of the circle.
In general |x – c| > r consists of the numbers outside of the
"circle” centered at c with radius r,
i.e. {x ≤ c – r} U {c + r ≤ x}. c c+rc–r
r r
2–3= –1 2+3=5
2
right 3left 3or {x ≤ –1 } U {5 ≤ x}.
0 r–r
|x| > r:

38. I. (Two–Piece | |–Inequalities)
If |x| > r then the solution
are {x < –r} U {x < r}.
Absolute Value Inequalities
|x – 2| ≥ 3
the distance between x and 2 is at least 3
Translate the symbols to a geometric description.
Example B. Translate the meaning of |x – 2| ≥ 3 and solve.
The expression |x – c| > r means the “the distance from x to c is
more than r”. They are the two flanks outside of the circle.
In general |x – c| > r consists of the numbers outside of the
"circle” centered at c with radius r,
i.e. {x ≤ c – r} U {c + r ≤ x}. c c+rc–r
r r
2–3= –1 2+3=5
2
right 3left 3or {x ≤ –1 } U {5 ≤ x}.
(–∞, c – r) (c + r, ∞)
0 r–r
|x| > r:

39. Absolute Value Inequalities
We may pull out the coefficient of the x.

40. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| =

41. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)|

42. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|

43. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)|

44. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|

45. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
We use this step to help us to extract the geometric
information.

46. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
We use this step to help us to extract the geometric
information.

47. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
We use this step to help us to extract the geometric
information.
We want |2x + 3| = |2(x + 3/2)|

48. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
We use this step to help us to extract the geometric
information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3

49. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
We use this step to help us to extract the geometric
information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2
|x + 3/2| > 3/2

50. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
We use this step to help us to extract the geometric
information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2
|x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2

51. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2
We use this step to help us to extract the geometric
information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2
|x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2

52. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2 more than 3/2
We use this step to help us to extract the geometric
information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2
|x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2

53. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2 more than 3/2
–3/2
We use this step to help us to extract the geometric
information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2
|x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2

54. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2 more than 3/2
–3/2
x x
right 3/2left 3/2
We use this step to help us to extract the geometric
information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2
|x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2

55. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2 more than 3/2
–3 0
–3/2
x x
right 3/2left 3/2
We use this step to help us to extract the geometric
information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2
|x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2

56. Absolute Value Inequalities
We may pull out the coefficient of the x.
For example,
|2x – 4| = |2(x – 2)| = 2|x – 2|
|2x + 3| = |2(x + 3/2)| = 2|x + 3/2|
Example D. Solve geometrically |2x + 3| > 3.
the distance between x and –3/2 more than 3/2
Hence x < –3 or 0 < x.
–3 0
–3/2
x x
right 3/2left 3/2
We use this step to help us to extract the geometric
information.
We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2
|x + 3/2| > 3/2 or
|x – (–3/2)| > 3/2

57. Absolute Value Inequalities
Example E. Express [2, 4] as an absolute value inequality in x.
2 40

58. Absolute Value Inequalities
To write an interval [a, b] into an absolute value inequality,
a b
Example E. Express [2, 4] as an absolute value inequality in x.
2 40

59. Absolute Value Inequalities
To write an interval [a, b] into an absolute value inequality,
view [a, b] as a “circle”
a b
Example E. Express [2, 4] as an absolute value inequality in x.
2 40

60. Absolute Value Inequalities
To write an interval [a, b] into an absolute value inequality,
view [a, b] as a “circle” with center at its midpoint m = (a + b)/2,
a bm = (b + a )/2
Example E. Express [2, 4] as an absolute value inequality in x.
2 40

61. Absolute Value Inequalities
Example E. Express [2, 4] as an absolute value inequality in x.
To write an interval [a, b] into an absolute value inequality,
view [a, b] as a “circle” with center at its midpoint m = (a + b)/2,
with radius r = (b – a)/2 which is half of the distance from a to b.
2 40
a b
r = (b – a) /2
m = (b + a )/2

62. Absolute Value Inequalities
Example E. Express [2, 4] as an absolute value inequality in x.
To write an interval [a, b] into an absolute value inequality,
view [a, b] as a “circle” with center at its midpoint m = (a + b)/2,
with radius r = (b – a)/2 which is half of the distance from a to b.
2 40
a b
r = (b – a) /2
m = (b + a )/2
l x – m l ≤ r
An abs-value inequality

63. Absolute Value Inequalities
Example E. Express [2, 4] as an absolute value inequality in x.
To write an interval [a, b] into an absolute value inequality,
view [a, b] as a “circle” with center at its midpoint m = (a + b)/2,
with radius r = (b – a)/2 which is half of the distance from a to b.
2 4m=30
a b
r = (b – a) /2
m = (b + a )/2
l x – m l ≤ r
Viewing [2, 4] as a circle, its center is at m = (2 + 4) / 2 = 3,
An abs-value inequality

64. Absolute Value Inequalities
Example E. Express [2, 4] as an absolute value inequality in x.
To write an interval [a, b] into an absolute value inequality,
view [a, b] as a “circle” with center at its midpoint m = (a + b)/2,
with radius r = (b – a)/2 which is half of the distance from a to b.
2 4m=30
a b
r = (b – a) /2
m = (b + a )/2
l x – m l ≤ r
Viewing [2, 4] as a circle, its center is at m = (2 + 4) / 2 = 3,
with radius |4 – 2| / 2 = 1
An abs-value inequality

65. Absolute Value Inequalities
Example E. Express [2, 4] as an absolute value inequality in x.
To write an interval [a, b] into an absolute value inequality,
view [a, b] as a “circle” with center at its midpoint m = (a + b)/2,
with radius r = (b – a)/2 which is half of the distance from a to b.
2 4m=30
a b
r = (b – a) /2
m = (b + a )/2
l x – m l ≤ r
Viewing [2, 4] as a circle, its center is at m = (2 + 4) / 2 = 3,
with radius |4 – 2| / 2 = 1 so the interval is lx – 3l ≤ 1.
An abs-value inequality

66. Ex. A. Translate and solve the expressions geometrically.
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. |2x + 1| ≥ 3
21. |5 – 2x| ≤ 3 22. |3 + 2x| < 7 23. |–2x + 3| > 5
24. |4 – 2x| ≤ –3 25. |3x + 3| < 5 26. |3x + 1| ≤ 7
Absolute Value Inequalities
15. |x – 2| < 1 16. |3 – x| ≤ 5 17. |x – 5| < 5
18. |7 – x| < 3 19. |8 + x| < 9 20. |x + 1| < 3
Ex. B. Divide by the coefficient of the x–term, translate and
solve the expressions geometrically. Draw the solution.
27. |8 – 4x| ≤ 3 28. |4x – 5| < 9 29. |4x + 1| ≤ 9

67. Ex. C. Express the following intervals as absolute value
inequalities in x.
30. [–5, 5]
Absolute Value Inequalities
31. (–5, 5) 32. (–5, 2) 33. [–2, 5]
34. [7, 17] 35. (–49, 84) 36. (–11.8, –1.6) 37. [–1.2, 5.6]