# 3 7 absolute inequalities-algebraic(optional)

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## 3 7 absolute inequalities-algebraic(optional)Presentation Transcript

• More on Absolute Value Inequalities
(Algebraic Method–Optional)
• 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”.
• 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”.
Example B. Draw the inequality |x| < 7.
• 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”.
Example B. Draw the inequality |x| < 7.
We are to draw all numbers which are within 7 units from the number 0.
• 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”.
Example B. Draw the inequality |x| < 7.
We are to draw all numbers which are within 7 units from the number 0.
x
x
0
-7
-7
7
• 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”.
Example B. Draw the inequality |x| < 7.
We are to draw all numbers which are within 7 units from the number 0.
x
x
0
-7
-7
7
-7 < x < 7
• 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”.
Example B. Draw the inequality |x| < 7.
We are to draw all numbers which are within 7 units from the number 0.
x
x
0
-7
-7
7
-7 < x < 7
The open circles means the end points are not included in the solution.
• 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”.
Example B. Draw the inequality |x| < 7.
We are to draw all numbers which are within 7 units from the number 0.
x
x
0
-7
-7
7
-7 < x < 7
The open circles means the end points are not included in the solution.
I. (one piece | |–inequalities)
• 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”.
Example B. Draw the inequality |x| < 7.
We are to draw all numbers which are within 7 units from the number 0.
x
x
0
-7
-7
7
-7 < x < 7
The open circles means the end points are not included in the solution.
I. (one piece | |–inequalities)
If |x| < c then –c < x < c.
• 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”.
Example B. Draw the inequality |x| < 7.
We are to draw all numbers which are within 7 units from the number 0.
x
x
0
-7
-7
7
-7 < x < 7
The open circles means the end points are not included in the solution.
I. (one piece | |–inequalities)
If |x| < c then –c < x < c.
In general, if we have
|expression| < c
• 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”.
Example B. Draw the inequality |x| < 7.
We are to draw all numbers which are within 7 units from the number 0.
x
x
0
-7
-7
7
-7 < x < 7
The open circles means the end points are not included in the solution.
I. (one piece | |–inequalities)
If |x| < c then –c < x < c.
In general, if we have
|expression| < c
we rewrite it without the "| |" as
– c < expression < c.
• More on Absolute Value Inequalities
Example C. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
• Example C. Rewrite by dropping the "| |“, don’t solve.
a. | x + y | < 2
–2 < x + y < 2
• More on Absolute Value Inequalities
Example C. 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
Example C. 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
Example C. 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 D. Solve the inequality |3 – 2x| < 7 and draw the solution.
• More on Absolute Value Inequalities
Example C. 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 D. 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
Example C. 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 D. 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
Example C. 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 D. 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
Example C. 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 D. 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
Example C. 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 D. 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
Example C. 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 D. 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
Example C. 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 D. 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
0
-2
5
• More on Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is more than c”.
• More on Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is more than c”.
Example E. 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”.
Example E. Draw the inequality |x| > 7.
We are to draw all x’s which are 7 or more units from the number 0.
• More on Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is more than c”.
Example E. Draw the inequality |x| > 7.
We are to draw all x’s which are 7 or more units from the number 0.
-7
7
0
x < –7 or 7 < x
• More on Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is more than c”.
Example E. Draw the inequality |x| > 7.
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.
-7
7
0
x < –7 or 7 < x
• More on Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is more than c”.
Example E. Draw the inequality |x| > 7.
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.
-7
7
0
x < –7 or 7 < x
The solid circles means the end points are part of the solution.
• More on Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is more than c”.
Example E. Draw the inequality |x| > 7.
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.
-7
7
0
x < –7 or 7 < x
The solid circles means the end points are part of the solution.
II. (two–piece | |–inequalities)
• More on Absolute Value Inequalities
More on Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is more than c”.
Example E. Draw the inequality |x| > 7.
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.
-7
7
0
x < –7 or 7 < x
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
The expression |x| > c means “the distance from x to 0 is more than c”.
Example E. Draw the inequality |x| > 7.
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.
-7
7
0
x < –7 or 7 < x
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.
c < x
x< –c
-c
c
0
• More on Absolute Value Inequalities
The expression |x| > c means “the distance from x to 0 is more than c”.
Example E. Draw the inequality |x| > 7.
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.
-7
7
0
x < –7 or 7 < x
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.
c < x
x< –c
-c
c
0
In general, if we have the inequality
|expression| > c
we drop the | | and rewrite it as two inequalities
expression < – c or c < expression
• More on Absolute Value Inequalities
Example F. Rewrite without the "| |“, don’t solve
A. | x + y | > 2
• More on Absolute Value Inequalities
Example F. 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
Example F. 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
Example F. 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
Example F. 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 G. Solve the inequality |3x – 2| > 4. Draw.
• More on Absolute Value Inequalities
Example F. 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 G. 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
Example F. 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 G. 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
Example F. 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 G. 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
Example F. 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 G. 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
Example F. 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 G. 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
Example F. 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 G. 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
2
0
• More on Absolute Value Inequalities
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
15. |x – 2| < 1
16. |3 – x| ≤ 5
17. |x – 5| < 5
18. |7 – x| < 3
19. |1 – 2x| < 9
20. |2x + 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