How do you solve and write the following in interval notation: #10<=-2(x-2)<=20#?

1 Answer
Oct 8, 2017

Answer:

See a solution process below:

Explanation:

First, divide each segment of the system of inequalities by #color(blue)(-2)# to eliminate the need for parenthesis while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we must reverse the inequality operators:

#10/color(blue)(-2) color(red)(>=) (-2(x - 2))/color(blue)(-2) color(red)(>=) 20/color(blue)(-2)#

#-5 color(red)(>=) (color(blue)(cancel(color(black)(-2)))(x - 2))/cancel(color(blue)(-2)) color(red)(>=) -10#

#-5 color(red)(>=) x - 2 color(red)(>=) -10#

Now, add #color(red)(2)# to each segment to solve for #x# while keeping the system balanced:

#-5 + color(red)(2) >= x - 2 + color(red)(2) >= -10 + color(red)(2)#

#-3 >= x - 0 >= -8#

#-3 >= x >= -8#

Or

#x <= -3# and #x >= -8#

Or

#x >= -8# and #x <= -3#

Or, in interval notation:

#[-8, -3]#