Explain why solving an equation containing one absolute value expression does not apply to the expression | x - 5 | = -3?

One useful and naturally intuitive definition is that the absolute value of an number is it's distance from 0 on the number line. If we use this definition then we require that the absolute value of a real valued quantity be greater than or equal to zero. We also know that both sides of any equation must have the same sign to have a solution- therefore this equation is a contradiction.

can you simplify x^2 over absolute value of x?

Please help me solve this equation : -1 - |-2(2)|

-1 - |-2(2)|
is the same as... 


 
Notice that the absolute value "strips off" negative signs, but the result is still subtracted from -1 

:)

Solve for v.|4v-5|=|4v-7|

4v-5=4v-7 <-- no solution

or...

4v-5=-4v+7
8v=12
v=3/2

|2a+5|=|3a-3|

four cases but only 2 are unique: 
1) 2a+5=3a-3 or 
2) 2a+5+-3a+3 


so that 

case 1: 
a=8 
or 

case 2:
5a=-2
a=-2/5

Solve: |2x+1|=|3x-1|

|2x+1|=|3x-1| 

means 
3x-1 = 2x+1 or 3x-1= -(2x+1) 

which tells us 
x=2 or 5x=0 (thus x=0) 

solutions: {0, 2}

|x-1| = 5x+10

absolute value means...
1) x-1=5x+10 
or 
2) x-1=-5x-10 

so for case 1)...
4x=-11
x=-11/4 (extraneous)


and for case 2)...
-6x=9
x=-3/2

Find the real numbers that satisfy the equation: |x| = -1/5

Absolute values are always greater than or equal to 0, so no real number(s) can make this equation true (no solution).

Solve -8|2+2x|+4=20

 

 


 

 

Impossible absolute values are never negative 

No solution possible 

:)

Solve |8n+9|+6=1

 

 

 


Impossible absolute values can never be negative 

No solution 

:)

Solve |2x+5|=3

 

means 

2x+5=-3 or 2x+5=3
2x=-8 2x=-2
x=-4 x=-1 


:)

Solve: |x+5|=3x-7

|x+5|=3x-7 
means 
x+5=3x-7 or -(x+5)=3x-7
2x=12 4x=2 
x=6 x=1/2 

clearly x=1/2 is not a solution(why?), but x=6 is a solution 



:)

the absolute value of one less than a number is equal to one more than a number

the absolute value of one less than a number:  
is equal to : =

one more than a number: x+1 


this situation is clearly impossible for any number other than 0 


:)

solve |x+3|-|5-x|=|x+2|

For convenience we write: |x+3|=|x+2|+|x-5| 

We'll solve by checking the following cases: 
+ + +
+ + -
+ - +
+ - - 



+++
x+3=x+2+x-5
x=6 (yes-solution) 

++-
x+3=x+2-(x-5)
x=4 (yes-solution) 

+-+
x+3=-(x+2)+x-5
x=-10 (no-not a solution) 

+--
x+3=-(x+2)-(x-5)
x=0 (no-not a solution) 

solution {4, 6} 

:)

express and evaluate the distance between 84 and -34 using absolute value

the distance between any two points a and b on the number line can be written as 
 

Therefore 

 


:)

Solve 3|2x+5|=9x-6

After dividing all terms by 3 we get 
 

which means 

 
 
 
but this is not a solution! (why?) 
or... 

 
 



:)

Write the expression without absolute value symbols. |2x| - |x-1]\| ,given 0 < x < 1 The brackets are absolute value systems

since x is known to be positive we do not need the absolute value brackets and we write: 
2x -(x-1)=
x+1

What Does This Mean |-9| & What is Equivalent to it ?

|-9| means "what is the distance from 0 to -9 
and it is equal to 9

solve: 5|6-5x|=15x-35

 
 
 or  



 or  
neither of which works :( 

no solution!

Solve: 1/4|4x+7|=8x+16

 
 
 
 or  
 or  (extraneous!)