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1-4 Study Guide and InterventionSolving Absolute Value Equations

Absolute Value Expressions The absolute value of a number is its distance from 0 on a number line. The symb

| x| is used to represent the absolute value of a number x.

AbsoluteValue

•Words For any real number a, if a is positive or zero, te absolute value of a is a! If a is ne"ative, te absolute value

of a is te opposite of a!

•Symbols For any real number a, |a| # a, if a $ %, an& |a| # 'a, if a ( %!

Example 1: Evaluate |−4|   – |−2 x| if x = 6.

|−4|  – |−2 x|  = |−4|  – |−2•6|

= |−4|  – |−12|

= 4 – 12

= –8

Example 2: Evaluate |2 x−3  y|  if x = –4 and y =

3.

|2 x−3  y|  = |2(−4)−3(3)|

= |−8−9|

= |−17|= 17

)apter * 24 Glencoe Algebra

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Exercises

Evaluate eac expression if w = –4! x = 2! y =1

2! and z = –6.

1. |2 x – 8| 2. |6  z | – | –7| 3. ! |w  z |

4. | x !| – |2w| ". | x| – | y| – | z | 6. |7 – x|  |" x|

#. |w – 4 x| $. |wz | – | xy| %. | z | – " |! yz |

1&. ! |w|  2 | z – 2 y| 11. | z | – 4 |2 z  y| 12. 10 – | xw|

13. |6 y  z |  | yz | 14. "|wx| 1

4  |4 x 8 y| 1". 7| yz | – "0

16. 14 – 2|w – xy| 1#. |2 x – y|  ! y 1$. | xyz |  |wxz |

1%. z | z |  x|  x| 2&. 12 – |10 x – 10 y| 21.1

2  |! z 8w|

22. | yz – 4w# – w 23.3

4

#wz # 1

2

#8 y# 24. xz – # xz #

1-4 Study Guide and Intervention (continued)

Solving Absolute Value Equations

Absolute Value E'uations $se the definition of absolute value to solve e%uations containin& absolute value

e'pressions.

For any real numbers a an& b, +ere b $ %, if a # b ten a # b or a # 'b!

(l)ays chec* your ans)ers by substitutin& them into the ori&inal e%uation. +ometimes computed solutions are not actua

solutions.

Example: (olve |2 x−3|  = 1#. )ec* +our

solutions.

)ase 1  a = b

2 x – " = 17

2 x – " " = 17 "

2 x = 20

 x = 10

),E)- |2 x – "| = 17

)apter * 25 Glencoe Algebra

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|2,10- – "| ≟ 17

|20 – "| ≟ 17

#17# ≟ 17

17 = 17 ✓

)ase 2  a = – b

2 x – " = –17

2 x – " " = –17 "

2 x = –14

 x = –7

),E)- |2 x – "| = 17

#2,–7- – "# ≟ 17

#–14 – "# ≟ 17

#–17# ≟ 17

17 = 17 ✓

)apter * 25 Glencoe Algebra

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There are t)o solutions 10 and –7.

Exercises

(olve eac e'uation. )ec* +our solutions.

1. # x 1!# = "7 2. #t – 4# – ! = 0

3. # x – !# = 4! 4. #m "# = 12 – 2m

". #!b /# 16 = 2 6. #1! – 2k # = 4!

#. !n 24 = #8 – "n# $. #8 !a# = 14 – a

%.1

3#4 p – 11# = p 4 1&. #" x – 1# = 2 x 11

11. |13  x+3|  = –1 12. 40 – 4 x = 2#" x – 10#

13. ! f – #" f 4# = 20 14. #4b "# = 1! – 2b

1".1

2#6 – 2 x# = " x 1 16. #16 – " x# = 4 x – 12