How to solve -3|x-1|+5=8

Question

absolute value

Answer 1

Hi Shane!

 

When solving absolute value equations, you must get the absolute value expression by itself first. Anything that is outside of the absolute value must be moved over to the other side of the equation. 

 

1) -3|x - 1| +5 - 5 = 8 - 5

    (-3|x - 1|) / -3 = 3/-3

 |x - 1| = -1

 

Now that we have the absolute value expression by itself, you can now use the rule of solving for absolute value: 

           |x| = a  set x = a AND x = -a and solve for each to get your two answers.

 

2) x - 1 = -1                                                    

    x - 1 + 1 = -1 + 1

   x = 0

 

AND

 

x - 1 = 1

x - 1 + 1 = 1 + 1

x = 2

 

Finally, ALWAYS check your answers. Absolute values are a tricky thing in that you can get an answer sometimes BUT when you plug in your answer to check, it doesn't work out (because of the absolute value sign). (Really you should be checking your answer for ALL math problems to double check your work :)

 

3) x = 0

-3|0-1| + 5

-3|-1| + 5

-3(1) +5

-3 + 5

2 ≠ 8 

Therefore x = 0 IS NOT a solution (had we not had the absolute value sign, we would have done -3 * -1 to get 3 which would have made this true; this is why absolute values are tricky!!)

 

x = 2

-3|2-1| + 5

-3|1| + 5

-3 + 5

2 ≠ 8

 

So the answer for this problem is NO SOLUTION! 

 

Now here is something to remember when you start to get used to working with these type problems: ONLY AFTER you have gotten the absolute value expression by itself, take a look at the equation. Remember, when we take the absolute value of ANY number, it is always positive. So once we got this equation by itself it was |x - 1| = -1. No matter what you put in for x, will it ever equal -1?? No!! Now please don't get this confused with just ANY equation equal to a negative number. The absolute value expression has to be by itself first before you can determine this. For example |x + 4| - 6 = -2. Just because it is equal to -2 does NOT mean it has no solution because the absolute value expression is not by itself. Once we add 6 to both sides, it is now |x + 4| = 4. Notice we now have a positive 4 on the right side. So we can solve this. 

Only use this rule once you get comfortable solving absolute value equations. And also don't get it confused with the answer of x. Once you solve for x, you can have a positive or negative number. X itself can be negative, just not the absolute value expression can not be set equal to a negative (before applying the rule of x = a and x = -a)

 

Please let me know if you need more explanation!