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

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

### Solving MATH problems

We can solve all math problems. Get help on the web or with our math app

Scroll to top