Download §3-3 ABSOLUTE VALUE EQUATIONS

§3-3
ABSOLUTE VALUE EQUATIONS
Theorem
If u = c where u is a variable expression and c is a non-negative real number,
then u = c or u = -c.
Example 1
Solve the equation for x: x = 6 .
Solution
x = 6 or x = -6
Example 2
Solve the equation for x: x + 3 = 5.
x+3=5
Solution
or
x + 3 = -5
x=2
Example 3
Solve the equation for y: 2 y − 4 = 7.
2y - 4 = 7
2y = 11
Solution
y=
Example 4
x = -8
or
2y - 4 = -7
2y = -3
11
2
y= −
3
2
Solve the equation for x: 3x + 2 − 13 = 7 .
3x + 2 − 13 = 7
3x + 2 = 20
Solution
3x + 2 = 20
3x = 18
or
3x + 2 = -20
3x = -22
x= −
x=6
Example 5
Solution
22
3
Solve the equation for t: t 2 + 2 = 7 .
t +2 =7
2
t =5
t2 = ± 5
2
t= ± 5
or
t + 2 = -7
2
t = -9
t 2 = ± −9
2
t = ± 3i
So t = ± 5 are the only real solutions.
Copyright©2007 by Lawrence Perez and Patrick Quigley
§3-3
PROBLEM SET
Solve each equation. If there is no solution then write no solution.
1.
x =5
2.
x =0
4.
x +3= 5
5.
p −
1 1
=
2 4
5
2
3.
3v =
6.
x = −2
7.
3 t = 12
8.
3t = 12
9.
y
=5
3
10.
1
w =4
3
11.
x −4 = 2
12.
5x + 1 = 10
13.
3a − a = 3
14.
1
2x + 1 = 15
2
15.
− 2x + 1 = 8
16.
7z + 1 = 2
17.
4 3a − 1 − 4 = −3
18.
2
x + 4 = 16
3
19.
1 + 3q =
1
5
20.
3x + 1 + 6 = 10
21.
3x − 1 = 4
22.
3x + 2 − 4 = 4
23.
5 4x − 3 + 7 = 52
24.
1
t =4
2
25.
3n + 4 =
28.
1
2
26.
4x 2 − 9 = 0
27.
y−5 = 3
− 6y − 3 = 4
29.
3x − 2
=1
5
30.
x − 2 = 5x
31.
3t − 1 = t − 4
32.
4
1
r +9 =
5
3
33.
3 h − 14 − 7 = 2
34.
−2
37.
y−3 = 5
36.
2 x+2 = 4
4−
b
=3
2
39.
2x + 3 = 11
4s − 3
= −1
5
2
37.
2p = p − 4
38.
40.
2− t−5 =4
41.
8 − 1− 3s = −1
42.
1
3a − 7 + 8 = 10
3
44.
a x +b=y
45.
3x + y = 4
Solve for x.
43.
ax + b = y
Copyright©2007 by Lawrence Perez and Patrick Quigley
§3-3
PROBLEM SOLUTIONS
1.
{ -5, 5 }
2.
{0}
3.
⎧ 5, 5 ⎫
⎬
⎨−
⎩ 6 6 ⎭
4.
{ -2, 2 }
5.
⎧ 3, 3 ⎫
⎬
⎨−
⎩ 4 4 ⎭
6.
no solution
7.
{ -4, 4 }
8.
{ -4, 4 }
9.
{ -15, 15 }
10.
{ -12, 12 }
11.
{ 2, 6 }
12.
⎧ − 11 , 9 ⎫
⎨
⎬
⎩ 5 5⎭
13.
⎧ 3, 3 ⎫
⎨−
⎬
⎩ 2 2 ⎭
14.
⎧ − 31 , 29 ⎫
⎬
⎨
⎩ 2 2 ⎭
15.
⎧ − 7, 9 ⎫
⎨
⎬
⎩ 2 2 ⎭
16.
⎧ 3, 1 ⎫
⎬
⎨−
⎩ 7 7 ⎭
17.
⎧ 1, 5 ⎫
⎨
⎬
⎩ 4 12 ⎭
18.
{ -30, 18 }
19.
⎧ 4 ,− 2 ⎫
⎨−
⎬
⎩ 15 5 ⎭
20.
⎧ − 5, 1 ⎫
⎬
⎨
⎩ 3 ⎭
21.
⎧ − 1, 5 ⎫
⎬
⎨
⎩
3 ⎭
22.
⎧ 10 , 2 ⎫
⎨−
⎬
⎩ 3
⎭
23.
⎧ − 3,3 ⎫
⎬
⎨
⎩ 2 ⎭
24.
{ -8, 8 }
25.
⎧ 7, − 3 ⎫
⎬
⎨−
⎩ 6
2 ⎭
26.
⎧±3 ⎫
⎬
⎨
⎩ 2 ⎭
27.
{ 2, 8 }
28.
⎧ 7, 1 ⎫
⎬
⎨−
⎩ 6 6 ⎭
29.
⎧ − 1, 7 ⎫
⎬
⎨
⎩
3⎭
30.
⎧ 5 ± 33 −5 ± 33 ⎫
,
⎨
⎬
2
2
⎩
⎭
31.
⎧ 3, 5 ⎫
⎬
⎨−
⎩ 2 4 ⎭
32.
⎧ − 65 , − 35 ⎫
⎬
⎨
⎩ 6
3 ⎭
33.
{ 11, 17 }
34.
⎧ 1 , 11 ⎫
⎬
⎨
⎩8 8 ⎭
37.
{ -2, 8 }
36.
{ -4, 0 }
37.
4 ⎫
⎧
⎨ − 4, ⎬
⎩
3 ⎭
38.
{ 2, 14 }
39.
{ -7, 4 }
40.
no solution
41.
⎧ 8 10 ⎫
⎬
⎨− ,
⎩ 3 3 ⎭
42.
⎧ 1 13 ⎫
⎬
⎨ ,
⎩ 3 3 ⎭
43.
⎧ y +b, y −b ⎫
⎬
⎨−
⎩
a ⎭
a
44.
⎧ ± y−b ⎫
⎬
⎨
a ⎭
⎩
45.
⎧ − 4 + y, 4− y ⎫
⎬
⎨
⎩
3
3 ⎭
Copyright©2007 by Lawrence Perez and Patrick Quigley