How do you solve #4abs[x+1]-2<10#?

1 Answer
Nov 2, 2017

Answer:

See a solution process below:

Explanation:

First, add #color(red)(2)# to each side of the inequality to isolate the absolute value term while keeping the inequality balanced:

#4abs(x + 1) - 2 + color(red)(2) < 10 + color(red)(2)#

#4abs(x + 1) - 0 < 12#

#4abs(x + 1) < 12#

Now, divide each side of the inequality by #color(red)(4)# to isolate the absolute value function while keeping the inequality balanced:

#(4abs(x + 1))/color(red)(4) < 12/color(red)(4)#

#(color(red)(cancel(color(black)(4)))abs(x + 1))/cancel(color(red)(4)) < 3#

#abs(x + 1) < 3#

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

#-3 < x + 1 < 3#

Now, subtract #color(red)(1)# from each segment of the system of inequalities to solve for #x# while keeping the system balanced:

#-3 - color(red)(1) < x + 1 - color(red)(1) < 3 - color(red)(1)#

#-4 < x + 0 < 2#

#-4 < x < 2#

Or

#x > -4# and #x < 2#

Or, in interval notation:

#(-4, 2)#