What is the solution set for the equation #|4a + 6| − 4a = 10# ?
1 Answer
Explanation:
The first thing to do here is isolate the modulus on onse side of the equation by adding
#|4a + 6| - color(red)(cancel(color(black)(4a))) + color(red)(cancel(color(black)(4a))) = 10 + 4a#
#|4a + 6| = 10 + 4a#
Now, by definition, the absolute value of a real number will only return positive values, regardless of the sign of said number.
This means that the first condition that any value of
#10 + 4a >= 0#
#4a >= -10 implies a >= -5/2#
Keep this in mind. Now, since the absolute value of a number returns a positive value, you can have two possibilities
#4a + 6 < 0 implies |4a + 6| = -(4a+6)#
In this case, the equation becomes
#-(4a + 6) = 10 + 4a#
#-4a - 6 = 10 + 4a#
#8a = - 16 implies a = ((-16))/8 = -2#
#(4a + 6) >=0 implies |4a + 6| = 4a + 6#
This time, the equation becomes
#color(red)(cancel(color(black)(4a))) + 6 = 10 + color(red)(cancel(color(black)(4a)))#
# 6 != 10 implies a in O/#
Therefore, the only valid solution will be
Do a quick check to make sure that the calculations are correct
#|4 * (-2) + 6| - 4 * (-2) = 10#
#|-2| +8 = 10#
#2 + 8 = 10 color(white)(x)color(green)(sqrt())#