How do you use interval notation to express the solution to #7- ( x + 1) \geq 6- 9x#?

1 Answer
smendyka
May 23, 2018

See a solution process below:

Explanation:

First, expand and combine the terms on the left side of the inequality being careful to manage the signs of each term correctly:

#7 - x - 1 >= 6 - 9x#

#7 - 1 - x >= 6 - 9x#

#6 - x >= 6 - 9x#

Next, subtract #color(red)(6)# and add #color(blue)(9x)# to each side of the inequality to isolate the #x# term while keeping the inequality balanced:

#6 - color(red)(6) - x + color(blue)(9x) >= 6 - color(red)(6) - 9x + color(blue)(9x)#

#0 - x + color(blue)(9x) >= 0 - 0#

#-x + color(blue)(9x) >= 0#

#-1x + color(blue)(9x) >= 0#

#(-1 + color(blue)(9))x >= 0#

#8x >= 0#

Now, divide each side of the inequality by #color(red)(8)# to solve for #x# while keeping the equation balanced:

#(8x)/color(red)(8) >= 0/color(red)(8)#

#(color(red)(cancel(color(black)(8)))x)/cancel(color(red)(8)) >= 0#

#x >= 0#

Or, in interval notation:

#[0, +oo)#

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