|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 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

Add a and a.

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

Add 4 and 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 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

Add a and a.

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

Add 2 and 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