How do you solve and write the following in interval notation: #10

Question

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

Eva Abramson · Accepted Answer

See a solution process below:

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# Instead, using interval notation: #[-8, -3]#

Julia Adams · Answer

undefined To solve the inequality -2(x - 2) between 10 and 20, first, divide each part of the inequality by -2. This reverses the inequality signs. So, it becomes:

-5 <= x - 2 <= -10

Then, add 2 to each part:

-5 + 2 <= x <= -10 + 2

-3 <= x <= -8

Finally, write the interval notation:

[-3, -8]

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

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