How do you solve #|3b - 3| \leq 12#?

Question

Savannah Anderson · Accepted Answer

#-3<=b<=5#

Consider these condition:

#|-13|=+13 >12#

From this scenario we can deduce you may not have any value of #3b-3# that is less than #-12#

In the same way we may not have any value of #3b-3# that is greater than +12

Consider these points as limiting the range such that we have:

#-12<=3b-3<=12#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider the lower bound (-12)

Set #" "3b-3=-12color(white)("ddd") =>color(white)("ddd") b=(3-12)/3=-3#

So #color(white)("dd")3b-3>=-12color(white)("ddd")=>color(white)("dddd")b>=(3-12)/3#

#color(white)("ddddddddddddddddddd")=>color(white)("dddd")b>=-3#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider the upper bound (+12)

Set#color(white)("d")3b-3=+12color(white)("dddd")=>color(white)("dddd")b=(12+3)/3=+5#

So#color(white)("d")3b-3=<=12color(white)("dddd")=>color(white)("dddd")b<=(12+3)/3#

#color(white)("ddddddddddddddddddd")=>color(white)("dddd")b<=+5#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#-3<=b<=5#

Jameson Allen · Answer

undefined To solve the inequality $|3b - 3| \leq 12$, you first isolate the absolute value expression. Then, you split the inequality into two cases based on whether the expression inside the absolute value is positive or negative.

Case 1: $3b - 3$ is non-negative (i.e., $3b - 3 \geq 0$):
- Solve $3b - 3 \geq 0$ for $b$.
- Once you find the solution for this case, check if it satisfies the original inequality $|3b - 3| \leq 12$.

Case 2: $3b - 3$ is negative (i.e., $3b - 3 < 0$):
- Solve $3b - 3 < 0$ for $b$.
- Once you find the solution for this case, check if it satisfies the original inequality $|3b - 3| \leq 12$.

Combine the solutions from both cases to find the overall solution set for $b$.