Write as an equivalent absolute value equation.
x-4=4x-5 or x-4=-(4x-5)
x-4=abs(4x-5)
To write an equivalent equation using absolute value, we can use the fact that |a| = a or |a| = -a, depending on the sign of a.
For the first equation, x-4=4x-5, let's isolate the absolute value expression on one side. Subtracting x from both sides gives -4=3x-5. Adding 5 to both sides gives 1=3x. Dividing by 3, we find x=1/3.
Using the fact that |a| = a if a is positive, we can rewrite the equation as:
x-4=|4x-5|
For the second equation, x-4=-(4x-5), we first distribute the negative sign to the expression within the parentheses: x-4=-4x+5.
Now, let's isolate the absolute value expression on one side. Adding 4x to both sides gives us 5x-4=5. Adding 4 to both sides gives 5x=9. Finally, dividing both sides by 5 yields x=9/5.
Using the fact that |a| = -a if a is negative, we can rewrite the equation as:
x-4=-|4x-5|
Therefore, the absolute value equations are:
x-4=|4x-5| and x-4=-|4x-5|.