Question 230867
Equation is |3-4x| - 3 <= 8


This means that if the expression within the absolute value sign is positive, then:


3-4x - 3 <= 8


Combine like terms to get:


-4x <= 8


Divide both sides by 4 to get:


-x <= 2


Multiply both sides by -1 to get:


x >= -2


Note that the inequality is reversed when you multiply both sides by -1.


If the expression within the absolute value sign is negative, then:


-(3-4x) - 3 <= 8


This becomes:


-3 + 4x - 3 <= 8


Combine like terms to get:


-6 + 4x <= 8


Add 6 to both sides of the equation to get:


4x <= 8+6 becomes 4x <= 14


Divide both sides of the equation by 4 to get:


x <= 14/4


You have:


x >= -2


and:


x <= 14/4


This is equivalent to -2 <= x <= 14/4


We need to confirm these values are accurate.


We'll take values inside and outside the limits to see what happens.


We'll do:


x = -3 (outside limits)
x = -2 (within limits)
x = 3 (within limits)
x = 14/4 (within limits)
x = 4 (outside limits)


When x = -3, |3-4x| - 3 = |3+12| - 3 = |15| - 3 = 15-3 = 12 > 8 so this is NOT ok as it should be.


When x = -2, |3-4x| - 3 = |3+8| - 3 = |11| - 3 = 11 - 3 = 8 <= 8 so this IS ok as it should be.


When x = 3, |3-4x| - 3 = |3-12| - 3 = |-9| - 3 = 9 - 3 = 6 <= 8 so this IS ok as it should be.


When x = 14/4, |3-4x| - 3 = |3 - 14| - 3 = |-11| - 3 = 11 - 3 = 8 <= 8 so this IS ok as it should be.


When x = 4, |3-4x| - 3 = |3 - 16| - 3 = |-13| - 3 = 13 - 3 = 10 > 8 so this is NOT ok as it should be.


Looks likes these values are good.