How do you solve #5 - 2 | 3x - 4 | = -5#?

Answer 1

See the entire solution process below:

First, subtract #color(red)(5)# from each side of the equation to isolate the absolute value term and keep the equation balanced:
#5 - 2abs(3x - 4) - color(red)(5) = -5 - color(red)(5)#
#5 - color(red)(5) - 2abs(3x - 4) = -10#
#0 - 2abs(3x - 4) = -10#
#-2abs(3x - 4) = -10#
Next, divide each side of the equation by #color(red)(-2)# to solve for the absolute value term while keeping the equation balanced:
#(-2abs(3x - 4))/color(red)(-2) = (-10)/color(red)(-2)#
#(color(red)(cancel(color(black)(-2)))abs(3x - 4))/cancel(color(red)(-2)) = 5#
#abs(3x - 4) = 5#

Because the absolute value is a unique function that requires two solutions—it converts a positive or negative term to its positive form—the term inside the absolute value needs to be solved for both the positive and negative terms that it is equivalent to.

First Solution

#3x - 4 = -5#
#3x - 4 + 4 = -5 + 4#
#3x - 0 = -1#
#3x = -1#
#(3x)/3 = -1/3#
#(color(red)(cancel(color(black)(3)))x)/color(red)(cancel(color(black)(3))) = -1/3#
#x = -1/3#

Option 2)

#3x - 4 = 5#
#3x - 4 + 4 = 5 + 4#
#3x - 0 = 9#
#3x = 9#
#(3x)/3 = 9/3#
#(color(red)(cancel(color(black)(3)))x)/color(red)(cancel(color(black)(3))) = 3#
#x = 3#

This issue can be resolved by:

#x = -1/3# and #x = 3#
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Answer 2

To solve the equation (5 - 2 |3x - 4| = -5), follow these steps:

  1. First, isolate the absolute value term by adding 5 to both sides: (5 - 2|3x - 4| + 5 = 0).

  2. Simplify: (10 - 2|3x - 4| = 0).

  3. Divide both sides by -2: (\frac{10}{-2} - \frac{2|3x - 4|}{-2} = 0).

  4. Simplify: (-5 + |3x - 4| = 0).

  5. Move the absolute value term to the other side by adding 5: (|3x - 4| = 5).

  6. Now, you'll have two cases to consider:

    • Case 1: (3x - 4 = 5) when (3x - 4) is positive.
    • Case 2: (3x - 4 = -5) when (3x - 4) is negative.
  7. Solve each case separately:

    • Case 1: (3x - 4 = 5)
      • Add 4 to both sides: (3x = 9).
      • Divide both sides by 3: (x = 3).
    • Case 2: (3x - 4 = -5)
      • Add 4 to both sides: (3x = -1).
      • Divide both sides by 3: (x = -\frac{1}{3}).

So, the solutions are (x = 3) and (x = -\frac{1}{3}).

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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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