How do you solve and write the following in interval notation: # 3 abs(4-5x)<=9#?

1 Answer
Jan 9, 2018

Answer:

See a solution process below: #[1/5, 7/5]#

Explanation:

First, divide each side of the inequality by #color(red)(3)# to isolate the absolute value term while keeping the inequality balanced:

#(3abs(4 - 5x))/color(red)(3) <= 9/color(red)(3)#

#(color(red)(cancel(color(black)(3)))abs(4 - 5x))/cancel(color(red)(3)) <= 3#

#abs(4 - 5x) <= 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 <= 4 - 5x <= 3#

Next, subtract #color(red)(4)# from each segment of the system of inequalities to isolate the #x# term while keeping the system balanced:

#-3 - color(red)(4) <= 4 - color(red)(4) - 5x <= 3 - color(red)(4)#

#-7 <= 0 - 5x <= -1#

#-7 <= -5x <= -1#

Now, divide each segment by #color(blue)(-5)# to solve for #x# while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we must reverse the inequality operators:

#(-7)/color(blue)(-5) color(red)(>=) (-5x)/color(blue)(-5) color(red)(>=) (-1)/color(blue)(-5)#

#7/5 color(red)(>=) (color(blue)(cancel(color(black)(-5)))x)/cancel(color(blue)(-5)) color(red)(>=) 1/5#

#7/5 color(red)(>=) x color(red)(>=) 1/5#

Or

#x >= 1/5#; #x <= 7/5#

Or, in interval notation

#[1/5, 7/5]#