3 3 absolute inequalities-geom-x

Absolute Value Inequalities (Geometric Method)

Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method.

Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method.

Example A. Translate the meaning of |x| < 7 and draw x.

Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method. We interpret these absolute-value inequalities as statements about distances then obtain the solutions by taking measurements on the real line.


Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method. We interpret these absolute-value inequalities as statements about distances then obtain the solutions by taking measurements on the real line. |x| means “the distance between x and 0”,



Absolute Value Inequalities In this section we solve simple absolute–value inequalities by the geometric method. We interpret these absolute-value inequalities as statements about distances then obtain the solutions by taking measurements on the real line. |x| means “the distance between x and 0”, so the expression |x| < c means “the distance between x and 0 is smaller than c” (provided that c is not negative in which case no such x exists).



|x| < 7

the distance between x and 0



|x| < 7

the distance between x and 0 is less than 7.



These are all the numbers which are within 7 units from 0, from –7 to 7.

|x| means “the distance between x and 0”, so the expression |x| < c means “the distance between x and 0 is smaller than c” (provided that c is not negative in which case no such x exists).

|x| < 7



–7 < x < 7-7-7 70

x



x


|x| < 7



–7 < x < 7-7-7 70

x



x

The open circles means the end points are not included in the solution.


|x| < 7


I. (One Piece | |–Inequalities) If |x| < c then –c < x < c.

Absolute Value Inequalities



Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”.



Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to solve these problems by simple drawings.




Example B. Translate the meaning of |x – 2| < 3 and solve.




|x – 2| < 3

Translate the symbols to a geometric description.





|x – 2| < 3

the distance between x and 2






|x – 2| < 3

the distance between x and 2 less than 3






|x – 2| < 3


2


Draw





|x – 2| < 3


2

x xright 3left 3


Draw





|x – 2| < 3


–1 52

x xright 3left 3


Draw





|x – 2| < 3


Hence –1 < x < 5.–1 52

x xright 3left 3


Draw


Absolute Value Inequalities I. (Two Piece | |–Inequalities) If |x| > c then x < –c or c < x.


Example C. Translate the meaning of |x| ≥ 7 and draw.

I. (Two Piece | |–Inequalities) If |x| > c then x < –c or c < x.


The geometric meaning of the inequality is that “the distance between x and 0 is 7 or more”.




The geometric meaning of the inequality is that “the distance between x and 0 is 7 or more”. In picture


-7-7 70x x


end point included




x < –7 or 7 < x-7-7 70x x


end point included



Note that the answer uses the word “or” for expressing the choices of two sections–not the word “and”.


x < –7 or 7 < x-7-7 70x x


end point included



Note that the answer uses the word “or” for expressing the choices of two sections–not the word “and”. The word “and” is used when there are multiple conditions to be met.


x < –7 or 7 < x-7-7 70x x


end point included





x < –7 or 7 < x-7-7 70x x


Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| > c means the “the distance between x and y is more than c”.

end point included





x < –7 or 7 < x-7-7 70x x


Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| > c means the “the distance between x and y is more than c”. We recall that we may break up | | for multiplication i.e. |x * y| = |x| * |y|.

end point included

Absolute Value Inequalities We use this fact to pull out the coefficient of the x.

Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| =

Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)|

Absolute Value Inequalities We use this fact to pull out the coefficient of the x. For example, |2x – 4| = |2(x – 2)| = 2|x – 2|

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

Absolute Value Inequalities We use this fact to 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|

Absolute Value Inequalities We use this fact to 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.


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)|




We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3




We want |2x + 3| = |2(x + 3/2)| = 2|x + 3/2| > 3 div. by 2 |x + 3/2| > 3/2




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



the distance between x and –3/2



|x – (–3/2)| > 3/2



the distance between x and –3/2 more than 3/2



|x – (–3/2)| > 3/2




–3/2



|x – (–3/2)| > 3/2




–3/2

x xright 3/2left 3/2



|x – (–3/2)| > 3/2




–3 0–3/2




|x – (–3/2)| > 3/2




Hence x < –3 or 0 < x.–3 0–3/2




|x – (–3/2)| > 3/2

Absolute Value Inequalities Let’s express intervals as absolute value inequalities.

I. (One Piece | |–Inequalities)


I. (One Piece | |–Inequalities) If –c < x < c then |x| < c, i.e.




–c 0right cleft c

+c|x| < c

Let’s express intervals as absolute value inequalities.



–c 0right cleft c

Recall that |x – y| translates into “the distance between x and y”

+c|x| < c



–c 0right cleft c

Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”.

+c|x| < c



–c 0right cleft c

Recall that |x – y| translates into “the distance between x and y” so the expression |x – y| < c means the “the distance between x and y is less than c”. We use this geometric meaning to write intervals into inequalities. Specifically, we need to locate the midpoint of the given interval first.

+c|x| < c



–c 0right cleft c


+c|x| < c

The mid-point m between two numbers x and y

In picture: o bm



–c 0right cleft c


+c|x| < c

The mid-point m between two numbers x and y is their average

that is m = . a + b2 In picture:

o b(a+b)/2

m



–c 0right cleft c


+c|x| < c

The mid-point m between two numbers x and y is their average

that is m = . a + b2

For example, the mid-point of 2 and 4 is (2 + 4)/2 = 3.

In picture: o b

(a+b)/2m

2 4(2+4)/2m = 30


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

Absolute Value Inequalities To write an interval [a, b] into an absolute value inequality,

a b


Absolute Value Inequalities To write an interval [a, b] into an absolute value inequality, view [a, b] as a “circle”

a b


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.

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




2 40

a b

r = (b – a) /2

m = (b + a )/2l x – m l ≤ r

An abs-value inequality




2 4m=30

a b

r = (b – a) /2

m = (b + a )/2l x – m l ≤ r

Viewing [2, 4] as a circle, its center is at m = (2 + 4) / 2 = 3,





2 4m=30

a b

r = (b – a) /2

m = (b + a )/2l 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





2 4m=30

a b

r = (b – a) /2

m = (b + a )/2l 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.


Il. (Two–Piece | |–Inequalities)



If x < –c or c < x then c < |x| (c > 0), i.e.





–c 0right cleft c

+c




–c 0right cleft c

+cc < |x|





–c 0right cleft c

+cc < |x|


Likewise the expression c < |x – y| means the “the distance between x and y is more than c”.




–c 0right cleft c

+cc < |x|


Example F. Express the following two intervals as an absolute value inequality in x.

2 40





–c 0right cleft c

+cc < |x|



2 40

Let’s find the abs-value inequality of the gap (2, 4).





–c 0right cleft c

+cc < |x|



2 4m=30

Let’s find the abs-value inequality of the gap (2, 4).The interval (2, 4) has its center at m = (2 + 4) / 2 = 3,with radius (4 – 2)/ 2 = 1





–c 0right cleft c

+cc < |x|



2 4m=30

Let’s find the abs-value inequality of the gap (2, 4).The interval (2, 4) has its center at m = (2 + 4) / 2 = 3,with radius (4 – 2)/ 2 = 1 so (2, 4) is lx – 3l < 1.


lx – 3l < 1




–c 0right cleft c

+cc < |x|



2 4m=30

Let’s find the abs-value inequality of the gap (2, 4).The interval (2, 4) has its center at m = (2 + 4) / 2 = 3,with radius (4 – 2)/ 2 = 1 so (2, 4) is lx – 3l < 1. So the intervals to the two sides is lx – 3l >1.


lx – 3l < 1

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


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

Ex. C. Express the following intervals as absolute value inequalities in x.

30. [–5, 5]


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]

–2 4

–15 –2

8 38

0 a

–a –a/2

a – b a + b

39.

42.

40.

41.

38.

43.

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3 3 absolute inequalities-geom-x