Solve for x |x-1|+|x-3|=2

$| x - 1 | + | x - 3 | = 2$

Step 1

Subtract $| x - 1 |$ from both sides of the equation.

$| x - 3 | = 2 - | x - 1 |$

Step 2

Solve for $| x - 1 |$ .

Rewrite the equation as $2 - | x - 1 | = | x - 3 |$ .

$2 - | x - 1 | = | x - 3 |$

Subtract $2$ from both sides of the equation.

$- | x - 1 | = | x - 3 | - 2$

Divide each term in $- | x - 1 | = | x - 3 | - 2$ by $- 1$ and simplify.

Divide each term in $- | x - 1 | = | x - 3 | - 2$ by $- 1$ .

$\frac{- | x - 1 |}{- 1} = \frac{| x - 3 |}{- 1} + \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{| x - 1 |}{1} = \frac{| x - 3 |}{- 1} + \frac{- 2}{- 1}$

Divide $| x - 1 |$ by $1$ .

$| x - 1 | = \frac{| x - 3 |}{- 1} + \frac{- 2}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{| x - 3 |}{- 1}$ .

$| x - 1 | = - 1 \cdot | x - 3 | + \frac{- 2}{- 1}$

Rewrite $- 1 \cdot | x - 3 |$ as $- | x - 3 |$ .

$| x - 1 | = - | x - 3 | + \frac{- 2}{- 1}$

Divide $- 2$ by $- 1$ .

$| x - 1 | = - | x - 3 | + 2$

Step 3

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

$x - 1 = \pm (- | x - 3 | + 2)$

Step 4

The result consists of both the positive and negative portions of the $\pm$ .

$x - 1 = - | x - 3 | + 2$

$x - 1 = - (- | x - 3 | + 2)$

Step 5

Solve $x - 1 = - | x - 3 | + 2$ for $x$ .

Solve for $| x - 3 |$ .

Rewrite the equation as $- | x - 3 | + 2 = x - 1$ .

$- | x - 3 | + 2 = x - 1$

Move all terms not containing $| x - 3 |$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$- | x - 3 | = x - 1 - 2$

Subtract $2$ from $- 1$ .

$- | x - 3 | = x - 3$

Divide each term in $- | x - 3 | = x - 3$ by $- 1$ and simplify.

Divide each term in $- | x - 3 | = x - 3$ by $- 1$ .

$\frac{- | x - 3 |}{- 1} = \frac{x}{- 1} + \frac{- 3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{| x - 3 |}{1} = \frac{x}{- 1} + \frac{- 3}{- 1}$

Divide $| x - 3 |$ by $1$ .

$| x - 3 | = \frac{x}{- 1} + \frac{- 3}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$| x - 3 | = - 1 \cdot x + \frac{- 3}{- 1}$

Rewrite $- 1 \cdot x$ as $- x$ .

$| x - 3 | = - x + \frac{- 3}{- 1}$

Divide $- 3$ by $- 1$ .

$| x - 3 | = - x + 3$

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

$x - 3 = \pm (- x + 3)$

The result consists of both the positive and negative portions of the $\pm$ .

$x - 3 = - x + 3$

$x - 3 = - (- x + 3)$

Solve $x - 3 = - x + 3$ for $x$ .

Move all terms containing $x$ to the left side of the equation.

Add $x$ to both sides of the equation.

$x - 3 + x = 3$

Add $x$ and $x$ .

$2 x - 3 = 3$

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

Add $3$ to both sides of the equation.

$2 x = 3 + 3$

Add $3$ and $3$ .

$2 x = 6$

Divide each term in $2 x = 6$ by $2$ and simplify.

Divide each term in $2 x = 6$ by $2$ .

$\frac{2 x}{2} = \frac{6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{6}{2}$

Divide $x$ by $1$ .

$x = \frac{6}{2}$

Simplify the right side.

Divide $6$ by $2$ .

$x = 3$

Solve $x - 3 = - (- x + 3)$ for $x$ .

Simplify $- (- x + 3)$ .

Rewrite.

$x - 3 = 0 + 0 - (- x + 3)$

Simplify by adding zeros.

$x - 3 = - (- x + 3)$

Apply the distributive property.

$x - 3 = - - x - 1 \cdot 3$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$x - 3 = 1 x - 1 \cdot 3$

Multiply $x$ by $1$ .

$x - 3 = x - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$x - 3 = x - 3$

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$x - 3 - x = - 3$

Combine the opposite terms in $x - 3 - x$ .

Subtract $x$ from $x$ .

$0 - 3 = - 3$

Subtract $3$ from $0$ .

$- 3 = - 3$

Since $- 3 = - 3$ , the equation will always be true.

All real numbers

Consolidate the solutions.

$x = 3$

Step 6

Solve $x - 1 = - (- | x - 3 | + 2)$ for $x$ .

Solve for $| x - 3 |$ .

Rewrite the equation as $- (- | x - 3 | + 2) = x - 1$ .

$- (- | x - 3 | + 2) = x - 1$

Simplify $- (- | x - 3 | + 2)$ .

Apply the distributive property.

$- - | x - 3 | - 1 \cdot 2 = x - 1$

Multiply $- - | x - 3 |$ .

Multiply $- 1$ by $- 1$ .

$1 | x - 3 | - 1 \cdot 2 = x - 1$

Multiply $| x - 3 |$ by $1$ .

$| x - 3 | - 1 \cdot 2 = x - 1$

Multiply $- 1$ by $2$ .

$| x - 3 | - 2 = x - 1$

Move all terms not containing $| x - 3 |$ to the right side of the equation.

Add $2$ to both sides of the equation.

$| x - 3 | = x - 1 + 2$

Add $- 1$ and $2$ .

$| x - 3 | = x + 1$

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

$x - 3 = \pm (x + 1)$

The result consists of both the positive and negative portions of the $\pm$ .

$x - 3 = x + 1$

$x - 3 = - (x + 1)$

Solve $x - 3 = x + 1$ for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$x - 3 - x = 1$

Combine the opposite terms in $x - 3 - x$ .

Subtract $x$ from $x$ .

$0 - 3 = 1$

Subtract $3$ from $0$ .

$- 3 = 1$

Since $- 3 \neq 1$ , there are no solutions.

No solution

Solve $x - 3 = - (x + 1)$ for $x$ .

Simplify $- (x + 1)$ .

Rewrite.

$x - 3 = 0 + 0 - (x + 1)$

Simplify by adding zeros.

$x - 3 = - (x + 1)$

Apply the distributive property.

$x - 3 = - x - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$x - 3 = - x - 1$

Move all terms containing $x$ to the left side of the equation.

Add $x$ to both sides of the equation.

$x - 3 + x = - 1$

Add $x$ and $x$ .

$2 x - 3 = - 1$

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

Add $3$ to both sides of the equation.

$2 x = - 1 + 3$

Add $- 1$ and $3$ .

$2 x = 2$

Divide each term in $2 x = 2$ by $2$ and simplify.

Divide each term in $2 x = 2$ by $2$ .

$\frac{2 x}{2} = \frac{2}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{2}{2}$

Divide $x$ by $1$ .

$x = \frac{2}{2}$

Simplify the right side.

Divide $2$ by $2$ .

$x = 1$

Consolidate the solutions.

$x = 1$

Step 7

Consolidate the solutions.

$x = 3, 1$

Step 8

Use each root to create test intervals.

$x < 1$

$1 < x < 3$

$x > 3$

Step 9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < 1$ to see if it makes the inequality true.

Choose a value on the interval $x < 1$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

$| (0) - 1 | + | (0) - 3 | = 2$

The left side $4$ does not equal to the right side $2$ , which means that the given statement is false.

False

Test a value on the interval $1 < x < 3$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 3$ and see if this value makes the original inequality true.

$x = 2$

Replace $x$ with $2$ in the original inequality.

$| (2) - 1 | + | (2) - 3 | = 2$

The left side $2$ is equal to the right side $2$ , which means that the given statement is always true.

True

Test a value on the interval $x > 3$ to see if it makes the inequality true.

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

$| (6) - 1 | + | (6) - 3 | = 2$

The left side $8$ does not equal to the right side $2$ , which means that the given statement is false.

False

Compare the intervals to determine which ones satisfy the original inequality.

$x < 1$ False

$1 < x < 3$ True

$x > 3$ False

$x < 1$ False

$1 < x < 3$ True

$x > 3$ False

Step 10

The solution consists of all of the true intervals.

$1 \leq x \leq 3$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$1 \leq x \leq 3$

Interval Notation:

$[1, 3]$

Step 12

Solve for x |x-1|+|x-3|=2

Related Questions

Support LCMGCF.com by disabling your adblocker.