Question

|3x-2|=19

Enter your question in the blanks in order from least to greatest.
3x-2=_ and 3x-2=_
3x-2=_____ and 3x-2=_____

Answer 1

The solutions to the absolute value equation |3x-2|=19 are x=7 and x=-17/3 or approximately -5.67. When placed in order from least to greatest, they are x=-5.67 and x=7. Both solutions are verified by substitution back into the original equation.

Step-by-step explanation:

The student is asking to solve the absolute value equation |3x-2|=19. To do this, we need to consider both the positive and negative cases since the absolute value of a number is its distance from zero on the number line and is always nonnegative.

Step-by-Step Solution:

First, we set up two separate equations to account for the positive and negative scenarios:

1. Positive case: If 3x-2 is positive or zero, then |3x-2|=3x-2. So we have 3x-2 = 19

2. Negative case: If 3x-2 is negative, then |3x-2|=-(3x-2), or 2-3x. So we have 2-3x = 19

Now we solve for x in each case:

1. 3x-2 = 19
   Add 2 to both sides: 3x = 21
   Divide by 3: x = 7

2. 2-3x = 19
   Subtract 2 from both sides: -3x = 17
   Divide by -3: x = -17/3 or approximately -5.67

So the solutions for the equation are x=7 and x=-17/3. Ordering them from least to greatest gives:

3x-2=-5.67 and 3x-2=7

Checking the Solutions:

After solving, we substitute these values back into the original equation to check if they are reasonable. For x=7, the substitution gives |3(7)-2| = |21-2| = |19| = 19, which checks out. For x=-17/3, the substitution gives |3(-17/3)-2| = |-17-2| = |-19| = 19, which also checks out.