Algebra 2. Absolute value problems help!?
If you are willing to solve these two for me, would you please show your work. I've worked them out and can't figure out where I'm going wrong.
Solve for x: | -2x + 3 | = | x - 15 |
6, -6
6, -12
-6, 12
-6 only
Solve for x: | x/6 + 5 | + 1/3 = 3/4
0, -5
-25, -30
-55/2, -65/2
0, -9
Comments
| -2x + 3 | = | x - 15 |
Square both sides of the equation to remove mod
4x^2 - 12x + 9 = x^2 - 30x +225
3x^2 + 18x - 216 = 0
x^2 + 6x - 72 = 0
Solving for x,
x = 6, -12
| x/6 + 5 | + 1/3 = 3/4
Multiply by 12,
| 2x + 60 | = 5
2x + 60 = 5 OR 2x + 60 = -5
(recall | x | = a means x = a OR x = -a )
For the two cases, solving for x,
x = -55/2 OR x = -65/2
Hence x = -55/2, -65/2
When dealing with absolute value equations, you take the absolute value on the left-hand side, remove it from the absolute value bars, and set it equal to first the right-hand side, then the negative right-hand side. In the case where the right-hand side is also an absolute value, simply remove it from the absolute value bars and proceed as normal.
-1-
|-2x + 3| = |x - 15|
-2x + 3 = x - 15 or -2x + 3 = -x + 15
-3x = -18 or -x = 12
x = 6 or x = -12
CHECK:
|-2(6)+3| = |6-15|
|-12+3| = |-9|
|-9| = |-9|
9 = 9 TRUE
|-2(-12)+3| = |-12-15|
|24+3| = |-27|
|27| = |-27|
27 = 27 TRUE
So your answer is x = 6 and x = -12
-2-
|x/6 + 5| + 1/3 = 3/4
First subtract 1/3 from both sides so that you have just the absolute value on the left-hand side.
|x/6 + 5| = 5/12
x/6 + 5 = 5/12 or x/6 + 5 = -5/12
x/6 = -55/12 or x/6 = -65/12
x = -55/2 or x = -65/2
CHECK
(-55/2)/6 + 5 = 5/12
-55/12 + 5 = 5/12
5/12 = 5/12 TRUE
(-65/2)/6 + 5 = -5/12
-65/12 + 5 = -5/12
-5/12 = -5/12 TRUE
So your answer is x = -55/2 and x = -65/2
The easiest way is to just plug in each answer and see if it works out. remember to apply the absolute values as the last step.
So, -2x+3 becomes -2(6)+3=-12+3=-9, which absolute value turns into plain 9. Since 6-15=-9, which absolute value makes plain 9. 6 is a valid answer for the equation.
I hope this makes sense. Good luck.
first one
I-2x+3I=Ix-15I..............squaring both side
4x^2+9-12x=x^2+225-30x
3x^2+18x-216=0
x^2+6x-72=0
(x+12)(x-6)=0
x=-12,6