How do you evaluate #7| 10+ v | - 1\leq 90#?

2 Answers
Sep 15, 2017

#-23le v le3#

Explanation:

#7|10+v|-1le90#
First, we will add #1# to both sides.
#7|10+v|-1color(blue)+color(blue)1le90color(blue)+color(blue)1#
This is also equal to
#7|10+v|le91#
Now we divide both sides by #7#
#(cancel7|10+v|)/cancel7lecancel91/cancel7 13#
This means that #|10+v|le13#.
And it also means that #10-vge-13#

#10+vle13#
Subtract #10# from both sides.
#10+vcolor(blue)-color(blue)10le13color(blue)-color(blue)10#
Simplify.
#vle3#

#10+vge-13#
Subtract #10# from both sides.
#10+vcolor(blue)-color(blue)10ge-13color(blue)-color(blue)10#
Simplify.
#vge-23#

Combine it together.
#-23levle3#

Sep 15, 2017

See a solution process below:

Explanation:

First, add #color(red)(1)# to each side of the inequality to isolate the absolute value term while keeping the inequality balanced:

#7abs(10 + v) - 1 + color(red)(1) <= 90 + color(red)(1)#

#7abs(10 + v) - 0 <= 91#

#7abs(10 + v) <= 91#

Next, divide each side of the inequality by #color(red)(7)# to isolate the absolute value function while keeping the inequality balanced:

#(7abs(10 + v))/color(red)(7) <= 91/color(red)(7)#

#(color(red)(cancel(color(black)(7)))abs(10 + v))/cancel(color(red)(7)) <= 13#

#abs(10 + v) <= 13#

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.

#-13 <= 10 + v <= 13#

Now, subtract #color(red)(10)# from each segment of the system of inequalities to solve for #v# while keeping the system balanced:

#-color(red)(10) - 13 <= -color(red)(10) + 10 + v <= -color(red)(10) + 13#

#-23 <= 0 + v <= 3#

#-23 <= v <= 3#

Or

#v >= -23# and #v <= 3#

Or, in interval notation:

#[-23, 3]#