Solving Absolute Value Equations and Inequalities

[Pages:6]Absolute Value Equations and Inequalities

Absolute Value Definition - The absolute value of x, is defined as...

|| =

-, ,

0 -8 - < -6 - > -10 > 6 < 10

(, )

h. 3|4 - 1| 9 |4 - 1| 3

i.

| + 6| > 0

Set + 6 0 So -6

(-, -) (-, )

Problem "h" can be solved using two different approaches.

Option 1 ? Split in to two different Inequalities joined by an "AND" statement (Intersection)

3|4 - 1| 9 |4 - 1| 3

4 - 1 3 4 - 1 -3

Option 2 ? Write as a compound inequality (Intersection)

3|4 - 1| 9 |4 - 1| 3

-3 4 - 1 3 (add 1) -2 4 4 (divide by 4)

1 -

- ,

- 1

- ,

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Steps for Solving NON- Linear Absolute Value Equations:

Follow the same steps as outlined for the linear absolute value equations, but all answers must be plugged back in to the original equation to verify whether they are valid or not (i.e. "Check your answers.") Occasionally, "extraneous" solutions can be introduced that are not correct and they must be excluded from the final answer.

Examples:

a. | + 1| = 5

2 Equations

+ 1 = 5 or + 1 = -5

x = 4

or = -6

= 4 or = -6

= ? or = !

Check your answers!

Check: = 2 |(2) + 1| = 5 |5| = 5

Check: = -2 |(-2) + 1 | = 5 |5| = 5

5=5

5=5

= Works!

= - Works!

b. | + 5 + 4| = 0 Only 1 Equation

+ 5 + 4 = 0 ( + 1)( + 4) = 0 + 1 = 0 and + 4 = 0 = - and = -

Check your answers!

Check: = -1

Check: = -4

|(-1) + 5(-1) + 4 | = 0 |(-4) + 5(-4) + 4 | = 0

|1 - 5 + 4| = 0

|16 - 20 + 4| = 0

|0| = 0 0 = 0

|0| = 0 0 = 0

= - Works!

= - Works!

c. | + 3| = - 4 - 3 2 Equations

+ 3 = - 4 - 3

or + 3 = -( - 4 - 3)

- 5 - 6 = 0 ( - 6)( + 1) = 0 - 6 = 0 and + 1 = 0 = and = -

or + 3 = - + 4 + 3 or - 3 = 0 or ( - 3) = 0 or = and =

Check your answers! Plugging each of the 4 answers into original equation results in ...

= -1 2 = 2 =6 9=9 = 0 3 -3 = 3 6 -6

So, the only answers to the problem are = - and = . ( = 0 = 3 are extraneous).

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Problem

1. || = 8

Absolute Value Practice Problems

Answer

-8, 8

Type

Equation, "+" number

2. | - 2| = 6 3. | + 1| = 0

-4, 8 -1

Equation, "+" number Equation, "zero"

4. | - 4| = -6

No Solution

Equation, "-" number

5. |3 + 2| = 10 6. |2 + 5| + 4 = 3

-4,

No Solution

Equation, "+" number Equation, "-" number

7. 2|4 - 1| = 6 8. |2 - 6| + 1 = 2

9. -3| - 1| - 6 = 3

- , 1

1, 5 No Solution

Equation, "+" number Equation, "+" number Equation, "-" number

10. | - 7| + 2 = 2

7

Equation, "zero"

11. |3 + 2| = | - 6|

-4, 1

Equation, | | = | |

12. | - 4| = |4 - |

Equation, | | = | |

13. || 2

-2 , 2

Inequality, "+" nmbr

14. | + 3| > 4

(- , -7) (-7 , ) Inequality, "+" nmbr

15. | + 3| < -6

No Solution

Inequality, "-" nmbr

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16. 3|2 - 4| -9 17. 2| - 9| + 6 > 6 18. -4|3 - 1| 8 19. -5|2 + 2| - 3 -3 20. -10 + | - 4| -10

21. 3 + 2 + 6 < 15

22. | - 4| = -4 23. | + 9 + 14| = 0 24. | + 1| = 2 25. | - 1| = + 4 - 5

(- , 9) (9 , ) No Solution -1 (-10 , 2) No Solution = -2, -7 =1 = -6, 1

Inequality, "-" nmbr Inequality, "zero" Inequality, "-" nmbr Inequality, "zero" Inequality, "zero" Inequality, "+" nmbr Equation, Non-Linear Equation, Non-Linear Equation, Non-Linear Equation, Non-Linear

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