(1.)
|x - 7| = 3
here is the problem
x - 7 = 3
x - 7 = -3
+
7 +7
__________
x = 10
(2.)
set equal to 3 and to
+7
+7
add 7 to each side
____________
x = 4
add
;
|3x + 2| = 5
here is the problem
3x + 2 = 5
3x + 2 = -5
-2 -2
__________
3x = 3
__ __
3
3
-2
-2 subtract 2 from each side
____________
3x = -7
subtract
__
___
3
3
divide each side by 3
;
x = 1
(3.)
-3
;
set equal to 5 and to -5
x = -7/3
cancel
|5x + 4| = -3
[no solution]
(4.)
|5 - 3x| = -2
[no solution]
(5.)
|2x - 6|
=
|4 - 5x|
2x - 6 = 4 - 5x
here is the problem
2x - 6 = -4 + 5x
+ 6 +6
+6 +6
_____________
__________________
2x = 10 - 5x ;
2x
= 2 + 5x
+ 5x
+ 5x
________________
7x = 10
;
___ ___
7
7
x = 10/7
add 6 to each side
add
-2x
-2x
add this to ea side
_________________
0 = 2 + 3x
add
___ ___ ___
3
3
3
divide thru by 3
;
0 = (2/3) + x
divide
____________
x = 10/3
results:
(6.)
-2/3
-2/3
_______________
-2/3 =
x = 10/3
x
subtract
x = -2/3
|2x - 1| = |4x + 3|
2x - 1 = 4x + 3
here is the problem
2x - 1 = -4x - 3
+ 1
+ 1
_______________
2x = 4x + 4
+1
+ 1 add 1 to ea side
___________________
2x
= -4x - 2
add
-4x
-4x
_________________
-2x =
4
____
__
2
2
+
x = -2
;
results: x = -2 ;
_____
(7.) √x - 3 = 4
4x
+ 4x
add this to ea side
__________________
6x
=
-2 add
___
___
6
6 div ea side by this
x = -1/3
dividez
x = -1/3
here is the problem
x - 3 = 16
square each side
+
3 + 3
______________
x = 19
______
(8.)
√3x + 1 = 5
add 3 to each side
add
here is the problem
3x + 1 = 25
-1
-1
_________________
3x
=
24
___
___
3
3
x = 8
__
_
subtract 2/3 fr ea side
__
square each side
subtract 1 from each side
subtract
divide each side by 3
divide and cancel
_
(9.)
√5x + √3 = √3x - √5
- √3
-√3
here is the problem
subtract √3 from each side
____________________________
√5x
=
√3x - √5 - √3
subtract
- √3x -√3x
subtract this from each side
____________________________
√5x - √3x =
-√5 - √3
subtract
√3x - √5x = √5 + √3
multiply thru by -1
3x - 2x√15 + 5x = √5 + √3
foil multiply
[and combine like terms]
8x - 2x√15 = √5 + √3
combine like terms on the left
x(8 - 2√15) = √5 + √3
___________ __________
8 - 2√15
8 - 2√15
factor
x = (√5 + √3)/(8 - 2√15)
(10.)
divide each side by this
cancel
_
_____
√x = 7 + √x - 7
here is the problem
x = 49 + 14√x - 7 + x - 7
-x
-x
______________________________
0 = 49 + 14√x - 7
- 7
0 = 49 + 14√x - 14
0 = 35 + 14√x
-35 -35
________________
square each side
subtract x from ea side
subtract
combine like terms
combine like terms
subtract 35 from each side
-35 =
14√x
subtract
result:
(11.)
no solution
__
_
√2x - 2 = 2x - √2
Let u = √2x
here is the problem
use this substitution
u - 2 = u2 - √2 make the substitution
+ 2
+ 2 add 2 to each side
____________________
u = u2 - √2 + 2
add
-u
- u
_______________________
0 = u2 - u - √2 + 2
b2 - 4ac
=
subtract u from each side
subtract
use the discriminant formula
(-1)2 - 4(1)(-√2 + 2)
make substitutions
=
1 + 4√2 - 8
multiply thru parentheses
=
4√2 - 7
combine like terms
=
-1.34314575
result:
use calculator
no solution
______
_____
(12.) √4x + 5 + 2√x - 3 = 17
here is the problem
______ _____
4x + 5 + 4√4x + 5√x - 3 + 4(x - 3) = 289 square each side
_______________
4x + 5 + 4√(4x + 5)(x - 3) + 4x - 12 = 289
multiply
_______________
8x - 7 + 4√(4x + 5)(x - 3) = 289 combine like terms
+ 7
+
7 add 7 to each side
_________________________________________________
8x
+ 4√(4x + 5)(x - 3) = 296
add
-8x
-8x
subtract 8x from each side
______________________________________
4√(4x + 5)(x - 3) = 296 - 8x
subtract
__________________ ____ ___
4
4
4 divide thru by 4
_______________
√(4x + 5)(x - 3) = 74 - 2x divide
(4x + 5)(x - 3) = 5476 - 296x + 4x2
4x2 - 7x - 15 = 5476 - 296x + 4x2
foil multiply
-4x2
-4x2 subtract from ea side
______________________________________
-7x - 15 = 5476 - 296x
subtract
+296x
+ 296x
add this to each side
_______________________________
289x - 15 = 5476
add
+ 15 +15
__________________
289x
= 5491
_____
______
289
289
x = 19
add 15 to each side
add
divide each side by 289
cancel
_____
_____
√x - 2 - √x + 3 = 1
here is the problem
_____
_____
x - 2 - 2(√x - 2)(√x + 3) + x + 3 = 1 square each side
_____
_____
2x + 1 - 2(√x - 2)(√x + 3) = 1
combine like terms
(13.)
-1
-1
subtract 1 from each side
_________________________________
2x
- 2(√x - 2)(√x + 3) = 0
subtract
x - (√x - 2)(√x + 3) = 0
-x + (√x - 2)(√x + 3) = 0
divide thru by 2, cancel
multiply thru by -1
+x
+
x
add x to each side
_____________________________
(√x - 2)(√x + 3)= x
add
(x - 2)(x + 3) = x2
square each side
x2 + x - 6 = x2
foil multiply combine like terms
-x2
-x2
subtract x2 from each side
___________________
x - 6 = 0
subtract
+ 6 +6
_____________
x = 6
NO SOLUTION
(1.)
add 6 to each side
add
[6 does not check out in the original problem]
|5 - 2x| > 3
here is the problem
5 - 2x > 3
5 - 2x < -3
-5 + 2x < -3
-5 + 2x > 3
multiply thru by -1
and change signs
+ 5
+ 5
+ 5
+ 5
_________________
2x < 2
;
2x > 8
___ __
__ __
2
2
2 2
x < 1
;
|(4/5)x - 1| > 3
(4/5)x - 1 > 3
4x - 5 > 15
+ 5 + 5
____________
add
divide each side by 2
x > 4
(-ininity, 1) U (4, infinity)
(2.)
add 5 to each side
divide and cancel
this is the interval notation
here is the problem
(4/5)x - 1 < -3
4x - 5 < -15
+ 5 +5
_______________
multiply thru by 5
add 5 to each side
4x
___
4
> 20
___
4
4x
___
4
x > 5
< -10
____
4
x < -2.5
|-2x + 6| > 8
divide ea side by 4
divide
(-ininity, -2.5) U (5, infinity)
(3.)
add
this is the interval notation
here is the problem
-2x + 6 > 8
-2x + 6 < -8
2x - 6 < -8
2x - 6 > 8
multiply thru by -1,
and change signs
+ 6
+6
________________
2x
< -2
__
__
2
2
x < -1
+6 +6
add 6 to each side
____________
2x
> 14
add
___
____
2
2
divide each side by 2
x > 7
divide and cancel
(-infinity, -1) U (7, infinity)
(4.)
|3x - 4|
>_ 2
3x - 4 >_ 2
this is the interval notation
here is the problem
3x - 4 <_ -2
+4 +4
+4
+ 4
add 4 to each side
____________
_______________
3x >_ 6
;
3x <_ 2
add
___
__
__
___
3
3
3
3
divide each side by 3
x >_ 2
;
x <_ 2/3
cancel
(-infinity, 2/3] U [2, infinity)
(5.)
|2x - 1| < 5
-5 < 2x - 1 < 5
this is the interval notation
here is the problem
+1
+ 1 + 1
add 1 to each side
___________________
-4 < 2x
< 6
add
___
___
___
2
2
2
divide thru by 2
-2 < x < 3
(-2,3)
(6.)
divide and cancel
this is the interval notation
|2x - 3| <_ 4
-4 <_
2x - 3 <_
here is the problem
4
+3
+ 3
+3
_____________________
-1 <_ 2x <_ 7
___
___
___
2
2
2
-1/2
<_
[-1/2, 7/2]
(7.)
x
<_
add 3 to each side
add
divide thru by 2
7/2
cancel
this is the interval notation
|3x - 1| <_ 8
-8 <_ 3x - 1 <_ 8
+ 1
+
1 + 1
___________________
-7
<_ 3x <_ 9
__
___
___
3
3
3
-7/3
<_
[-7/3, 3]
(8.)
x <_
3
add 1 to each side
add
divide thru by 3
divide and cancel
this is the interval notation
|(x/3) - 7| <_ 5
here is the problem
-5 <_ (x/3) - 7 <_ 5
-15 <_ x - 21 <_ 15
multiply thru by 3, cancel
+ 21 + 21
+ 21 add 21 to each side
________________________
6
<_
x
[6,36]
(9.)
<_
36
add
this is the interval notation
|(x/3) + 2| < 4
-4 < (x/3) + 2
here is the problem
< 4
-12 < x + 6 < 12
-6
-6
-6
___________________
-18
<
x
subtract 6 from each side
< 6
(-18, 6)
(10.)
multiply thru by 3
subtract
this is the interval notation
|(4/3) + x| <_ 2/5
here is the problem
-2/5 <_ (4/3) + x <_ 2/5
-6 <_ 20 + 15x <_ 6
multiply thru by 15, cancel
-20
-20
-20 subtract 20 from each side
________________________
-26 <_
15x <_ -14
subtract
___
____
____
15
15
15
divide thru by 15
-26/15
<_
[-26/15 ,
(11.)
x
<_ -14/15
-14/15]
this is the interval notation
|2x + 5| <_ x + 3
case 1:
cancel
here is the problem
2x + 5 <_ x + 3
-x
-x
subtract x from each side
__________________________
x + 5
<_
3
subtract
-5
-5
subtract 5 from each side
__________________________
x
case 2:
<_
-2
subtract
2x + 5 >_ -x - 3
+x
+x
add x to each side
________________________
3x + 5 >_
-3
add
-5
-5
subtract 5 from each side
_______________________
3x
>_
-8 subtract
____
___
3
3
divide each side by 3
x >_ -8/3
cancel
-8/3 <_ x <_ -2
[-8/3, -2]
this is the interval notation