How do you solve #abs(3x-9)-2<=7#?

Answer 1

#x in [0, 6]#

The first thing to do here is isolate the modulus on one side of the equation by adding #2# to both sides
#|3x-9| - color(red)(cancel(color(black)(2))) + color(red)(cancel(color(black)(2))) <= 7 + 2#
#|3x - 9| <= 9#

Now, you have two possible cases to deal with

In this case, the inequality becomes

#3x-9 <= 9#
#3x <= 18 implies x <= 18/3 = 6#

In this case, the inequality becomes

#-(3x-9) <= 9#
#-3x +9 <=9#
#-3x <= 0 implies x >= 0#
You thus know that any value of #x# that is smaller than or equal to #6# and greater than or equal to #0# will satisfy the original inequality. In interval notation, this is written as
#x in (-oo, 6] nn [0, + oo)#

This means that the solution interval will be

#x in [0, 6]#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To solve the inequality (|3x - 9| - 2 \leq 7):

  1. Add 2 to both sides: (|3x - 9| \leq 9).
  2. There are two cases to consider: a. (3x - 9 \leq 9) when (3x - 9) is positive. b. (3x - 9 \geq -9) when (3x - 9) is negative.
  3. Solve each case separately: a. For (3x - 9 \leq 9), add 9 to both sides: (3x \leq 18), then divide by 3: (x \leq 6). b. For (3x - 9 \geq -9), add 9 to both sides: (3x \geq 0), then divide by 3: (x \geq 0).
  4. Combine the solutions: (0 \leq x \leq 6).
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7