absolute value of x if x is positive and absolute value of x is -x if x is negative.
the absolute value of an expression is always positive.
your problem states that |-1/5 - 1/2 * k| = 9/5
by the definition, when the expression is positive it is equal to 9/5 and when the expression is negative, then minus the expression is equal to 9/5.
therefore:
when the expression is positive, you get:
(-1/5 - 1/2 * k) = 9/5
this becomes -1/5 - 1/2 * k = 9/5
add 1/5 to both sides of the equation and simplify to get:
-1/2 * k = 10/5
solve for k to get:
k = -20/5 = -4
confirm by replacing k in the original equation to get:
|-1/5 - 1/2 * k| = 9/5
this becomes |9/5| = 9/5 which becomes 9/5 = 9/5, confirming the value of k is correct.
when the expression is negative, you get:
-(-1/5 - 1/2 * k) = 9/5
this becomes 1/5 + 1/2 * k = 9/5
subtract 1/5 from both sides of the equation and simplify to get:
1/2 * k = 8/5
solve for k to get:
k = 16/5 = 3.2
confirm by replacing k in the original equation to get:
|-1/5 - 1/2 * 16/5| = 9/5 which becomes |-1/5 - 8/5| = 9/5 which becomes |-9/5| = 9/5 which becomes 9/5 = 9/5, confirming the value of k is good.
you can graph two equations as shown below:
y = |-1/5 - 1/2 * x| and y = 9/5
the intersection of the two equations is the solution.
that solution is x = -4 and x = 3.2
x = 3.2 is the same as x = 16/5.
you can also graph one equation as shown below:
y - |-1/5 - 1/2 * x| - y = 9/5
the intersection of the equation with the x-axis is the solution.
the graph of both solutions is shown below:
you can see that the x-coordinate of both graphs are the same.
those are your solutions.
the y-coordinates will be different because the second graph is shifted down 9/5 units of y.
y = 9/5 is the same as y = 1.8.
