# Solve for a |a-3|-|a-2|=1

|a-3|-|a-2|=1
Add |a-2| to both sides of the equation.
|a-3|=1+|a-2|
Reorder 1 and |a-2|.
|a-3|=|a-2|+1
Remove the absolute value term. This creates a ± on the right side of the equation because |x|=±x.
a-3=±(|a-2|+1)
The result consists of both the positive and negative portions of the ±.
a-3=|a-2|+1
a-3=-(|a-2|+1)
Solve a-3=|a-2|+1 for a.
Solve for |a-2|.
Rewrite the equation as |a-2|+1=a-3.
|a-2|+1=a-3
Move all terms not containing |a-2| to the right side of the equation.
Subtract 1 from both sides of the equation.
|a-2|=a-3-1
Subtract 1 from -3.
|a-2|=a-4
|a-2|=a-4
|a-2|=a-4
Remove the absolute value term. This creates a ± on the right side of the equation because |x|=±x.
a-2=±(a-4)
The result consists of both the positive and negative portions of the ±.
a-2=a-4
a-2=-(a-4)
Solve a-2=a-4 for a.
Move all terms containing a to the left side of the equation.
Subtract a from both sides of the equation.
a-2-a=-4
Combine the opposite terms in a-2-a.
Subtract a from a.
0-2=-4
Subtract 2 from 0.
-2=-4
-2=-4
-2=-4
Since -2≠-4, there are no solutions.
No solution
No solution
Solve a-2=-(a-4) for a.
Simplify -(a-4).
Apply the distributive property.
a-2=-a–4
Multiply -1 by -4.
a-2=-a+4
a-2=-a+4
Move all terms containing a to the left side of the equation.
Add a to both sides of the equation.
a-2+a=4
2a-2=4
2a-2=4
Move all terms not containing a to the right side of the equation.
Add 2 to both sides of the equation.
2a=4+2
2a=6
2a=6
Divide each term by 2 and simplify.
Divide each term in 2a=6 by 2.
2a2=62
Cancel the common factor of 2.
Cancel the common factor.
2a2=62
Divide a by 1.
a=62
a=62
Divide 6 by 2.
a=3
a=3
a=3
Consolidate the solutions.
a=3
a=3
Solve a-3=-(|a-2|+1) for a.
Solve for |a-2|.
Rewrite the equation as -(|a-2|+1)=a-3.
-(|a-2|+1)=a-3
Multiply each term in -(|a-2|+1)=a-3 by -1
Multiply each term in -(|a-2|+1)=a-3 by -1.
-(|a-2|+1)⋅-1=a⋅-1+(-3)⋅-1
Simplify -(|a-2|+1)⋅-1.
Apply the distributive property.
(-|a-2|-1⋅1)⋅-1=a⋅-1+(-3)⋅-1
Multiply -1 by 1.
(-|a-2|-1)⋅-1=a⋅-1+(-3)⋅-1
Apply the distributive property.
-|a-2|⋅-1-1⋅-1=a⋅-1+(-3)⋅-1
Multiply -|a-2|⋅-1.
Multiply -1 by -1.
1|a-2|-1⋅-1=a⋅-1+(-3)⋅-1
Multiply |a-2| by 1.
|a-2|-1⋅-1=a⋅-1+(-3)⋅-1
|a-2|-1⋅-1=a⋅-1+(-3)⋅-1
Multiply -1 by -1.
|a-2|+1=a⋅-1+(-3)⋅-1
|a-2|+1=a⋅-1+(-3)⋅-1
Simplify each term.
Move -1 to the left of a.
|a-2|+1=-1⋅a+(-3)⋅-1
Rewrite -1a as -a.
|a-2|+1=-a+(-3)⋅-1
Multiply -3 by -1.
|a-2|+1=-a+3
|a-2|+1=-a+3
|a-2|+1=-a+3
Move all terms not containing |a-2| to the right side of the equation.
Subtract 1 from both sides of the equation.
|a-2|=-a+3-1
Subtract 1 from 3.
|a-2|=-a+2
|a-2|=-a+2
|a-2|=-a+2
Remove the absolute value term. This creates a ± on the right side of the equation because |x|=±x.
a-2=±(-a+2)
The result consists of both the positive and negative portions of the ±.
a-2=-a+2
a-2=-(-a+2)
Solve a-2=-a+2 for a.
Move all terms containing a to the left side of the equation.
Add a to both sides of the equation.
a-2+a=2
2a-2=2
2a-2=2
Move all terms not containing a to the right side of the equation.
Add 2 to both sides of the equation.
2a=2+2
2a=4
2a=4
Divide each term by 2 and simplify.
Divide each term in 2a=4 by 2.
2a2=42
Cancel the common factor of 2.
Cancel the common factor.
2a2=42
Divide a by 1.
a=42
a=42
Divide 4 by 2.
a=2
a=2
a=2
Solve a-2=-(-a+2) for a.
Simplify -(-a+2).
Apply the distributive property.
a-2=–a-1⋅2
Multiply –a.
Multiply -1 by -1.
a-2=1a-1⋅2
Multiply a by 1.
a-2=a-1⋅2
a-2=a-1⋅2
Multiply -1 by 2.
a-2=a-2
a-2=a-2
Move all terms containing a to the left side of the equation.
Subtract a from both sides of the equation.
a-2-a=-2
Combine the opposite terms in a-2-a.
Subtract a from a.
0-2=-2
Subtract 2 from 0.
-2=-2
-2=-2
-2=-2
Since -2=-2, the equation will always be true.
All real numbers
All real numbers
Consolidate the solutions.
a=2
a=2
Consolidate the solutions.
a=3,2
Exclude the solutions that do not make |a-3|-|a-2|=1 true.
a=2
Solve for a |a-3|-|a-2|=1

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