3 4 absolute inequalities-algebraic-x

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”.

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



We are to draw all numbers which are within 7 units from the number 0.



-7-7 70

x



x



-7 < x < 7-7-7 70

x



x



-7 < x < 7-7-7 70

x



x

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


I. (One piece | |–inequalities)


-7 < x < 7-7-7 70

x



x



I. (One piece | |–inequalities) If |x| < c then –c < x < c.


-7 < x < 7-7-7 70

x



x





-7 < x < 7-7-7 70

x



x

In general, if we have |expression| < c





-7 < x < 7-7-7 70

x



x

In general, if we have |expression| < c we rewrite it without the "| |" as – c < expression < c.



Example B. Rewrite by dropping the "| |“, don’t solve.a. | x + y | < 2



–2 < x + y < 2



–2 < x + y < 2

b. | x2 – 2x + 1| < 1



–2 < x + y < 2

b. | x2 – 2x + 1| < 1

–1 <x2 – 2x + 1 < 1



–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.



–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



–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



–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


b. | x2 – 2x + 1| < 1


–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



–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



–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 < 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 50


The expression |x| > c means “the distance from x to 0 is more than c”.


Example D. Draw the inequality |x| > 7.





We are to draw all x’s which are 7 or more units from the number 0.




We are to draw all x’s which are 7 or more units from the number 0.

x < –7 or 7 < x-7 70




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





x < –7 or 7 < x-7 70

The solid circles means the end points are part of the solution.


II. (Two–piece | |–inequalities)




x < –7 or 7 < x-7 70

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”.


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.





x < –7 or 7 < x-7 70

-c c0

c < xx< –c






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



Example E. Rewrite without the "| |“, don’t solveA. | x + y | > 2


Example E. Rewrite without the "| |“, don’t solveA. | x + y | > 2 Drop the | |, write it as two inequalities asx + y < –2 or 2 < x + y B. | x2 – 2x + 1| > 1


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


3 4 absolute inequalities-algebraic-x

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