How do you solve and write the following in interval notation: #| -3x | + 2 ≤ 8#?

Answer 1

See the entire solution process below:

First, subtract #color(red)(2)# from each side of the inequality to isolate the absolute value function while keeping the inequality balanced:
#abs(-3x) + 2 - color(red)(2) <= 8 - color(red)(2)#
#abs(-3x) + 0 <= 6#
#abs(-3x) <= 6#

The absolute value function takes any positive or negative term and transforms it to its positive form. Therefore, you must solve the term within the absolute value function for both its positive and negative equivalent:

#-6 <= -3x <= 6#
We need to divide each segment of the system of inequalities by #color(blue)(-3)# to solve for #x# while keeping the system balanced. However, because we are multiplying or dividing an inequality by a negative term we must reverse the inequality operators:
#(-6)/color(blue)(-3) color(red)(>=) (-3x)/color(blue)(-3) color(red)(>=) 6/color(blue)(-3)#
#2 color(red)(>=) (color(blue)(cancel(color(black)(-3)))x)/cancel(color(blue)(-3)) color(red)(>=) -2#
#2 >= x >= -2#

Or

#x <= 2# and #x >= -2#

Or, in interval notation:

#[-2, 2]#
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Answer 2

To solve | -3x | + 2 ≤ 8 in interval notation, you would follow these steps:

  1. Subtract 2 from both sides: | -3x | ≤ 6
  2. Divide both sides by 3: | x | ≤ 2
  3. Split into two cases: x ≤ 2 and x ≥ -2
  4. Combine the two cases: -2 ≤ x ≤ 2

So, the solution in interval notation is [-2, 2].

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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.

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