Download 1-4 Study Guide and Intervention Solving Absolute Value Equations

NAME
1-4
DATE
PERIOD
Study Guide and Intervention
Solving Absolute Value Equations
Absolute Value Expressions The absolute value of a number is its distance from 0
on a number line. The symbol ⎪x⎥ is used to represent the absolute value of a number x.
• Words
For any real number a, if a is positive or zero, the absolute value of a is a.
If a is negative, the absolute value of a is the opposite of a.
• Symbols
For any real number a, ⎪a⎥ = a, if a ≥ 0, and ⎪a⎥ = -a, if a < 0.
Absolute Value
Example 1
if x = 6.
⎪-4⎥ - ⎪-2x⎥
Example 2
Evaluate 2x - 3y if
x = -4 and y = 3.
Evaluate -4 - -2x ⎪2x - 3y⎥ = ⎪2(-4) - 3(3)⎥
= ⎪-4⎥ - ⎪-2 6⎥
= ⎪-4⎥ - ⎪-12⎥
= 4 - 12
= -8
= ⎪-8 - 9⎥
= ⎪-17⎥
= 17
Exercises
1
Evaluate each expression if w = -4, x = 2, y = −
, and z = -6.
2
2. ⎪6 + z⎥ - ⎪-7⎥ -7
3. 5 + ⎪w + z⎥ 15
4. ⎪x + 5⎥ - ⎪2w⎥ -1
1
5. ⎪x⎥ - ⎪y⎥ - ⎪z⎥ -4 −
6. ⎪7 - x⎥ + ⎪3x⎥ 11
7. ⎪w - 4x⎥ 12
8. ⎪wz⎥ - ⎪xy⎥ 23
9. ⎪z⎥ - 3 ⎪5yz⎥ -39
2
10. 5 ⎪w⎥ + 2⎪z - 2y⎥ 34
11. ⎪z⎥ - 4 ⎪2z + y⎥ -40
12. 10 - ⎪xw⎥ 2
13. ⎪6y + z⎥ + ⎪yz⎥ 6
1
14. 3⎪wx⎥ + −
⎪4x + 8y⎥ 27
4
15. 7⎪yz⎥ - 30
16. 14 - 2⎪w - xy⎥ 4
17. ⎪2x - y⎥ + 5y 6
18. ⎪xyz⎥ + ⎪wxz⎥ 54
19. z⎪z⎥ + x⎪x⎥ -32
20. 12 - ⎪10x - 10y⎥ -3
1⎪
21. −
5z + 8w⎥ 31
2
22. ⎪yz - 4w⎥ -w 17
3⎪ ⎥
1
23. −
wz + −
⎪8y⎥ 20
4
2
24. xz - ⎪xz⎥
Chapter 1
24
-9
-24
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1. ⎪2x - 8⎥ 4
NAME
DATE
1-4
Study Guide and Intervention
PERIOD
(continued)
Solving Absolute Value Equations
Absolute Value Equations
Use the definition of absolute value to solve equations
containing absolute value expressions.
For any real numbers a and b, where b ≥ 0, if |a| = b then a = b or a = -b.
Always check your answers by substituting them into the original equation. Sometimes
computed solutions are not actual solutions.
Example
Solve 2x - 3 = 17. Check your solutions.
a = -b
2x - 3 = -17
2x - 3 + 3 = -17 + 3
2x = -14
x = -7
⎪2x - 3⎥ = 17
CHECK
|2(-7) - 3| 17
|-14 - 3| 17
|-17| 17
17 = 17 ✓
a=b
2x - 3 = 17
2x - 3 + 3 = 17 + 3
2x = 20
x = 10
⎪2x - 3⎥ = 17
CHECK
⎪2(10) - 3⎥ 17
⎪20 - 3⎥ 17
|17| 17
17 = 17 ✓
There are two solutions, 10 and -7.
Case 2
Exercises
Solve each equation. Check your solutions.
1. |x + 15| = 37
3. |x - 5| = 45
{-52, 22}
{-40, 50}
2. |t - 4| - 5 = 0
{-1, 9}
4. |m + 3| = 12 - 2m {3}
5. |5b + 9| + 16 = 2 ∅
6. |15 - 2k| = 45
7. 5n + 24 = |8 - 3n| {-2}
8. |8 + 5a| = 14 - a
1|
9. −
4p - 11| = p + 4
3
⎪
{23, - −17 }
⎥
{-15, 30}
10. |3x - 1| = 2x + 11
11
, 1}
{- −
2
{-2, 12}
1
11. −
x + 3 = -1 ∅
3
12. 40 - 4x = 2 |3x - 10|
13. 5f - |3f + 4| = 20 {12}
14. |4b + 3| = 15 - 2b
1|
15. −
6 - 2x| = 3x + 1
2
Chapter 1
{−12 }
{6, -10}
{2, -9}
16. |16 - 3x| = 4x - 12 {4}
25
Glencoe Algebra 2
Lesson 1-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Case 1