|a+b|=|a|+|b|

Reorder |a| and |b|.

|a+b|=|b|+|a|

Reorder |b| and |a|.

|a+b|=|a|+|b|

Remove the absolute value term. This creates a ± on the right side of the equation because |x|=±x.

a+b=±(|a|+|b|)

The result consists of both the positive and negative portions of the ±.

a+b=|a|+|b|

a+b=-(|a|+|b|)

Solve for |a|.

Rewrite the equation as |a|+|b|=a+b.

|a|+|b|=a+b

Subtract |b| from both sides of the equation.

|a|=a+b-|b|

|a|=a+b-|b|

Remove the absolute value term. This creates a ± on the right side of the equation because |x|=±x.

a=±(a+b-|b|)

The result consists of both the positive and negative portions of the ±.

a=a+b-|b|

a=-(a+b-|b|)

Solve a=a+b-|b| for a.

Solve for |b|.

Rewrite the equation as a+b-|b|=a.

a+b-|b|=a

Move all terms not containing |b| to the right side of the equation.

Subtract a from both sides of the equation.

b-|b|=a-a

Subtract b from both sides of the equation.

-|b|=a-a-b

Subtract a from a.

-|b|=0-b

Subtract b from 0.

-|b|=-b

-|b|=-b

Multiply each term in -|b|=-b by -1

Multiply each term in -|b|=-b by -1.

(-|b|)⋅-1=(-b)⋅-1

Multiply -|b|⋅-1.

Multiply -1 by -1.

1|b|=(-b)⋅-1

Multiply |b| by 1.

|b|=(-b)⋅-1

|b|=(-b)⋅-1

Multiply (-b)⋅-1.

Multiply -1 by -1.

|b|=1b

Multiply b by 1.

|b|=b

|b|=b

|b|=b

|b|=b

Remove the absolute value term. This creates a ± on the right side of the equation because |x|=±x.

b=±(b)

The result consists of both the positive and negative portions of the ±.

b=b

b=-(b)

Solve b=b for a.

Move all terms containing b to the left side of the equation.

Subtract b from both sides of the equation.

b-b=0

Subtract b from b.

0=0

0=0

Since 0=0, the equation will always be true.

Always true

Always true

Solve b=-(b) for a.

Multiply -1 by b.

b=-b

Move all terms containing b to the left side of the equation.

Add b to both sides of the equation.

b+b=0

Add b and b.

2b=0

2b=0

Divide each term by 2 and simplify.

Divide each term in 2b=0 by 2.

2b2=02

Cancel the common factor of 2.

Cancel the common factor.

2b2=02

Divide b by 1.

b=02

b=02

Divide 0 by 2.

b=0

b=0

b=0

Consolidate the solutions.

a=0

a=0

Solve a=-(a+b-|b|) for a.

Solve for |b|.

Rewrite the equation as -(a+b-|b|)=a.

-(a+b-|b|)=a

Multiply each term in -(a+b-|b|)=a by -1

Multiply each term in -(a+b-|b|)=a by -1.

-(a+b-|b|)⋅-1=a⋅-1

Simplify -(a+b-|b|)⋅-1.

Apply the distributive property.

(-a-b–|b|)⋅-1=a⋅-1

Multiply –|b|.

Multiply -1 by -1.

(-a-b+1|b|)⋅-1=a⋅-1

Multiply |b| by 1.

(-a-b+|b|)⋅-1=a⋅-1

(-a-b+|b|)⋅-1=a⋅-1

Apply the distributive property.

-a⋅-1-b⋅-1+|b|⋅-1=a⋅-1

Simplify.

Multiply -a⋅-1.

Multiply -1 by -1.

1a-b⋅-1+|b|⋅-1=a⋅-1

Multiply a by 1.

a-b⋅-1+|b|⋅-1=a⋅-1

a-b⋅-1+|b|⋅-1=a⋅-1

Multiply -b⋅-1.

Multiply -1 by -1.

a+1b+|b|⋅-1=a⋅-1

Multiply b by 1.

a+b+|b|⋅-1=a⋅-1

a+b+|b|⋅-1=a⋅-1

Move -1 to the left of |b|.

a+b-1⋅|b|=a⋅-1

a+b-1⋅|b|=a⋅-1

Rewrite -1|b| as -|b|.

a+b-|b|=a⋅-1

a+b-|b|=a⋅-1

Simplify a⋅-1.

Move -1 to the left of a.

a+b-|b|=-1⋅a

Rewrite -1a as -a.

a+b-|b|=-a

a+b-|b|=-a

a+b-|b|=-a

Move all terms not containing |b| to the right side of the equation.

Subtract a from both sides of the equation.

b-|b|=-a-a

Subtract b from both sides of the equation.

-|b|=-a-a-b

Subtract a from -a.

-|b|=-2a-b

-|b|=-2a-b

Multiply each term in -|b|=-2a-b by -1

Multiply each term in -|b|=-2a-b by -1.

(-|b|)⋅-1=(-2a)⋅-1+(-b)⋅-1

Multiply -|b|⋅-1.

Multiply -1 by -1.

1|b|=(-2a)⋅-1+(-b)⋅-1

Multiply |b| by 1.

|b|=(-2a)⋅-1+(-b)⋅-1

|b|=(-2a)⋅-1+(-b)⋅-1

Simplify each term.

Multiply -1 by -2.

|b|=2a+(-b)⋅-1

Multiply (-b)⋅-1.

Multiply -1 by -1.

|b|=2a+1b

Multiply b by 1.

|b|=2a+b

|b|=2a+b

|b|=2a+b

|b|=2a+b

|b|=2a+b

Remove the absolute value term. This creates a ± on the right side of the equation because |x|=±x.

b=±(2a+b)

The result consists of both the positive and negative portions of the ±.

b=2a+b

b=-(2a+b)

Solve b=2a+b for a.

Rewrite the equation as 2a+b=b.

2a+b=b

Move all terms not containing a to the right side of the equation.

Subtract b from both sides of the equation.

2a=b-b

Subtract b from b.

2a=0

2a=0

Divide each term by 2 and simplify.

Divide each term in 2a=0 by 2.

2a2=02

Cancel the common factor of 2.

Cancel the common factor.

2a2=02

Divide a by 1.

a=02

a=02

Divide 0 by 2.

a=0

a=0

a=0

Solve b=-(2a+b) for a.

Rewrite the equation as -(2a+b)=b.

-(2a+b)=b

Multiply each term in -(2a+b)=b by -1

Multiply each term in -(2a+b)=b by -1.

-(2a+b)⋅-1=b⋅-1

Simplify -(2a+b)⋅-1.

Apply the distributive property.

(-(2a)-b)⋅-1=b⋅-1

Multiply 2 by -1.

(-2a-b)⋅-1=b⋅-1

Apply the distributive property.

-2a⋅-1-b⋅-1=b⋅-1

Multiply -1 by -2.

2a-b⋅-1=b⋅-1

Multiply -b⋅-1.

Multiply -1 by -1.

2a+1b=b⋅-1

Multiply b by 1.

2a+b=b⋅-1

2a+b=b⋅-1

2a+b=b⋅-1

Simplify b⋅-1.

Move -1 to the left of b.

2a+b=-1⋅b

Rewrite -1b as -b.

2a+b=-b

2a+b=-b

2a+b=-b

Move all terms not containing a to the right side of the equation.

Subtract b from both sides of the equation.

2a=-b-b

Subtract b from -b.

2a=-2b

2a=-2b

Divide each term by 2 and simplify.

Divide each term in 2a=-2b by 2.

2a2=-2b2

Cancel the common factor of 2.

Cancel the common factor.

2a2=-2b2

Divide a by 1.

a=-2b2

a=-2b2

Cancel the common factor of -2 and 2.

Factor 2 out of -2b.

a=2(-b)2

Cancel the common factors.

Factor 2 out of 2.

a=2(-b)2(1)

Cancel the common factor.

a=2(-b)2⋅1

Rewrite the expression.

a=-b1

Divide -b by 1.

a=-b

a=-b

a=-b

a=-b

a=-b

Consolidate the solutions.

a=0

a=-b

a=0

a=-b

a=0

a=-b

Solve for |a|.

Rewrite the equation as -(|a|+|b|)=a+b.

-(|a|+|b|)=a+b

Multiply each term in -(|a|+|b|)=a+b by -1

Multiply each term in -(|a|+|b|)=a+b by -1.

-(|a|+|b|)⋅-1=a⋅-1+b⋅-1

Simplify -(|a|+|b|)⋅-1.

Apply the distributive property.

(-|a|-|b|)⋅-1=a⋅-1+b⋅-1

Apply the distributive property.

-|a|⋅-1-|b|⋅-1=a⋅-1+b⋅-1

Multiply -|a|⋅-1.

Multiply -1 by -1.

1|a|-|b|⋅-1=a⋅-1+b⋅-1

Multiply |a| by 1.

|a|-|b|⋅-1=a⋅-1+b⋅-1

|a|-|b|⋅-1=a⋅-1+b⋅-1

Multiply -|b|⋅-1.

Multiply -1 by -1.

|a|+1|b|=a⋅-1+b⋅-1

Multiply |b| by 1.

|a|+|b|=a⋅-1+b⋅-1

|a|+|b|=a⋅-1+b⋅-1

|a|+|b|=a⋅-1+b⋅-1

Simplify each term.

Move -1 to the left of a.

|a|+|b|=-1⋅a+b⋅-1

Rewrite -1a as -a.

|a|+|b|=-a+b⋅-1

Move -1 to the left of b.

|a|+|b|=-a-1⋅b

Rewrite -1b as -b.

|a|+|b|=-a-b

|a|+|b|=-a-b

|a|+|b|=-a-b

Subtract |b| from both sides of the equation.

|a|=-a-b-|b|

|a|=-a-b-|b|

Remove the absolute value term. This creates a ± on the right side of the equation because |x|=±x.

a=±(-a-b-|b|)

The result consists of both the positive and negative portions of the ±.

a=-a-b-|b|

a=-(-a-b-|b|)

Solve a=-a-b-|b| for a.

Solve for |b|.

Rewrite the equation as -a-b-|b|=a.

-a-b-|b|=a

Move all terms not containing |b| to the right side of the equation.

Add a to both sides of the equation.

-b-|b|=a+a

Add b to both sides of the equation.

-|b|=a+a+b

Add a and a.

-|b|=2a+b

-|b|=2a+b

Multiply each term in -|b|=2a+b by -1

Multiply each term in -|b|=2a+b by -1.

(-|b|)⋅-1=2a⋅-1+b⋅-1

Multiply -|b|⋅-1.

Multiply -1 by -1.

1|b|=2a⋅-1+b⋅-1

Multiply |b| by 1.

|b|=2a⋅-1+b⋅-1

|b|=2a⋅-1+b⋅-1

Simplify each term.

Multiply -1 by 2.

|b|=-2a+b⋅-1

Move -1 to the left of b.

|b|=-2a-1⋅b

Rewrite -1b as -b.

|b|=-2a-b

|b|=-2a-b

|b|=-2a-b

|b|=-2a-b

Remove the absolute value term. This creates a ± on the right side of the equation because |x|=±x.

b=±(-2a-b)

The result consists of both the positive and negative portions of the ±.

b=-2a-b

b=-(-2a-b)

Solve b=-2a-b for a.

Rewrite the equation as -2a-b=b.

-2a-b=b

Move all terms not containing a to the right side of the equation.

Add b to both sides of the equation.

-2a=b+b

Add b and b.

-2a=2b

-2a=2b

Divide each term by -2 and simplify.

Divide each term in -2a=2b by -2.

-2a-2=2b-2

Cancel the common factor of -2.

Cancel the common factor.

-2a-2=2b-2

Divide a by 1.

a=2b-2

a=2b-2

Simplify 2b-2.

Cancel the common factor of 2 and -2.

Factor 2 out of 2b.

a=2(b)-2

Move the negative one from the denominator of b-1.

a=-1⋅b

a=-1⋅b

Rewrite -1⋅b as -b.

a=-b

a=-b

a=-b

a=-b

Solve b=-(-2a-b) for a.

Rewrite the equation as -(-2a-b)=b.

-(-2a-b)=b

Multiply each term in -(-2a-b)=b by -1

Multiply each term in -(-2a-b)=b by -1.

-(-2a-b)⋅-1=b⋅-1

Simplify -(-2a-b)⋅-1.

Apply the distributive property.

(-(-2a)–b)⋅-1=b⋅-1

Multiply -2 by -1.

(2a–b)⋅-1=b⋅-1

Multiply –b.

Multiply -1 by -1.

(2a+1b)⋅-1=b⋅-1

Multiply b by 1.

(2a+b)⋅-1=b⋅-1

(2a+b)⋅-1=b⋅-1

Simplify by multiplying through.

Apply the distributive property.

2a⋅-1+b⋅-1=b⋅-1

Simplify the expression.

Multiply -1 by 2.

-2a+b⋅-1=b⋅-1

Move -1 to the left of b.

-2a-1⋅b=b⋅-1

-2a-1⋅b=b⋅-1

-2a-1⋅b=b⋅-1

Rewrite -1b as -b.

-2a-b=b⋅-1

-2a-b=b⋅-1

Simplify b⋅-1.

Move -1 to the left of b.

-2a-b=-1⋅b

Rewrite -1b as -b.

-2a-b=-b

-2a-b=-b

-2a-b=-b

Move all terms not containing a to the right side of the equation.

Add b to both sides of the equation.

-2a=-b+b

Add -b and b.

-2a=0

-2a=0

Divide each term by -2 and simplify.

Divide each term in -2a=0 by -2.

-2a-2=0-2

Cancel the common factor of -2.

Cancel the common factor.

-2a-2=0-2

Divide a by 1.

a=0-2

a=0-2

Divide 0 by -2.

a=0

a=0

a=0

Consolidate the solutions.

a=-b

a=0

a=-b

a=0

Solve a=-(-a-b-|b|) for a.

Solve for |b|.

Rewrite the equation as -(-a-b-|b|)=a.

-(-a-b-|b|)=a

Multiply each term in -(-a-b-|b|)=a by -1

Multiply each term in -(-a-b-|b|)=a by -1.

-(-a-b-|b|)⋅-1=a⋅-1

Simplify -(-a-b-|b|)⋅-1.

Apply the distributive property.

(–a–b–|b|)⋅-1=a⋅-1

Simplify.

Multiply –a.

Multiply -1 by -1.

(1a–b–|b|)⋅-1=a⋅-1

Multiply a by 1.

(a–b–|b|)⋅-1=a⋅-1

(a–b–|b|)⋅-1=a⋅-1

Multiply –b.

Multiply -1 by -1.

(a+1b–|b|)⋅-1=a⋅-1

Multiply b by 1.

(a+b–|b|)⋅-1=a⋅-1

(a+b–|b|)⋅-1=a⋅-1

Multiply –|b|.

Multiply -1 by -1.

(a+b+1|b|)⋅-1=a⋅-1

Multiply |b| by 1.

(a+b+|b|)⋅-1=a⋅-1

(a+b+|b|)⋅-1=a⋅-1

(a+b+|b|)⋅-1=a⋅-1

Apply the distributive property.

a⋅-1+b⋅-1+|b|⋅-1=a⋅-1

Simplify.

Move -1 to the left of a.

-1⋅a+b⋅-1+|b|⋅-1=a⋅-1

Move -1 to the left of b.

-1⋅a-1⋅b+|b|⋅-1=a⋅-1

Move -1 to the left of |b|.

-1⋅a-1⋅b-1⋅|b|=a⋅-1

-1⋅a-1⋅b-1⋅|b|=a⋅-1

Simplify each term.

Rewrite -1a as -a.

-a-1⋅b-1⋅|b|=a⋅-1

Rewrite -1b as -b.

-a-b-1⋅|b|=a⋅-1

Rewrite -1|b| as -|b|.

-a-b-|b|=a⋅-1

-a-b-|b|=a⋅-1

-a-b-|b|=a⋅-1

Simplify a⋅-1.

Move -1 to the left of a.

-a-b-|b|=-1⋅a

Rewrite -1a as -a.

-a-b-|b|=-a

-a-b-|b|=-a

-a-b-|b|=-a

Move all terms not containing |b| to the right side of the equation.

Add a to both sides of the equation.

-b-|b|=-a+a

Add b to both sides of the equation.

-|b|=-a+a+b

Add -a and a.

-|b|=0+b

Add 0 and b.

-|b|=b

-|b|=b

Multiply each term in -|b|=b by -1

Multiply each term in -|b|=b by -1.

(-|b|)⋅-1=b⋅-1

Multiply -|b|⋅-1.

Multiply -1 by -1.

1|b|=b⋅-1

Multiply |b| by 1.

|b|=b⋅-1

|b|=b⋅-1

Simplify b⋅-1.

Move -1 to the left of b.

|b|=-1⋅b

Rewrite -1b as -b.

|b|=-b

|b|=-b

|b|=-b

|b|=-b

Remove the absolute value term. This creates a ± on the right side of the equation because |x|=±x.

b=±(-b)

The result consists of both the positive and negative portions of the ±.

b=-b

b=-(-b)

Solve b=-b for a.

Move all terms containing b to the left side of the equation.

Add b to both sides of the equation.

b+b=0

Add b and b.

2b=0

2b=0

Divide each term by 2 and simplify.

Divide each term in 2b=0 by 2.

2b2=02

Cancel the common factor of 2.

Cancel the common factor.

2b2=02

Divide b by 1.

b=02

b=02

Divide 0 by 2.

b=0

b=0

b=0

Solve b=-(-b) for a.

Multiply -(-b).

Multiply -1 by -1.

b=1b

Multiply b by 1.

b=b

b=b

Move all terms containing b to the left side of the equation.

Subtract b from both sides of the equation.

b-b=0

Subtract b from b.

0=0

0=0

Since 0=0, the equation will always be true.

Always true

Always true

Consolidate the solutions.

a=0

a=0

Consolidate the solutions.

a=-b

a=0

a=-b

a=0

Consolidate the solutions.

a=0

a=-b

Solve for a |a+b|=|a|+|b|