Precalculus Examples

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

Step 1

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

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

Step 2

Solve for $| x - 1 |$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

Subtract $3$ from both sides of the equation.

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

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.2.2

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

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

Step 2.3.3

Simplify the right side.

Tap for more steps...

Step 2.3.3.1

Simplify each term.

Tap for more steps...

Step 2.3.3.1.1

Move the negative one from the denominator of $\frac{| x - 2 |}{- 1}$ .

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

Step 2.3.3.1.2

Rewrite $- 1 \cdot | x - 2 |$ as $- | x - 2 |$ .

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

Step 2.3.3.1.3

Divide $- 3$ by $- 1$ .

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

Step 3

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

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

Step 4

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

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

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

Step 5

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

Tap for more steps...

Step 5.1

Solve for $| x - 2 |$ .

Tap for more steps...

Step 5.1.1

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

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

Step 5.1.2

Move all terms not containing $| x - 2 |$ to the right side of the equation.

Tap for more steps...

Step 5.1.2.1

Subtract $3$ from both sides of the equation.

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

Step 5.1.2.2

Subtract $3$ from $- 1$ .

$- | x - 2 | = x - 4$

Step 5.1.3

Divide each term in $- | x - 2 | = x - 4$ by $- 1$ and simplify.

Tap for more steps...

Step 5.1.3.1

Divide each term in $- | x - 2 | = x - 4$ by $- 1$ .

$\frac{- | x - 2 |}{- 1} = \frac{x}{- 1} + \frac{- 4}{- 1}$

Step 5.1.3.2

Simplify the left side.

Tap for more steps...

Step 5.1.3.2.1

Dividing two negative values results in a positive value.

$\frac{| x - 2 |}{1} = \frac{x}{- 1} + \frac{- 4}{- 1}$

Step 5.1.3.2.2

Divide $| x - 2 |$ by $1$ .

$| x - 2 | = \frac{x}{- 1} + \frac{- 4}{- 1}$

Step 5.1.3.3

Simplify the right side.

Tap for more steps...

Step 5.1.3.3.1

Simplify each term.

Tap for more steps...

Step 5.1.3.3.1.1

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

$| x - 2 | = - 1 \cdot x + \frac{- 4}{- 1}$

Step 5.1.3.3.1.2

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

$| x - 2 | = - x + \frac{- 4}{- 1}$

Step 5.1.3.3.1.3

Divide $- 4$ by $- 1$ .

$| x - 2 | = - x + 4$

Step 5.2

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

$x - 2 = \pm (- x + 4)$

Step 5.3

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

$x - 2 = - x + 4$

$x - 2 = - (- x + 4)$

Step 5.4

Solve $x - 2 = - x + 4$ for $x$ .

Tap for more steps...

Step 5.4.1

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

Tap for more steps...

Step 5.4.1.1

Add $x$ to both sides of the equation.

$x - 2 + x = 4$

Step 5.4.1.2

Add $x$ and $x$ .

$2 x - 2 = 4$

Step 5.4.2

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

Tap for more steps...

Step 5.4.2.1

Add $2$ to both sides of the equation.

$2 x = 4 + 2$

Step 5.4.2.2

Add $4$ and $2$ .

$2 x = 6$

Step 5.4.3

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

Tap for more steps...

Step 5.4.3.1

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

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

Step 5.4.3.2

Simplify the left side.

Tap for more steps...

Step 5.4.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.4.3.2.1.1

Cancel the common factor.

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

Step 5.4.3.2.1.2

Divide $x$ by $1$ .

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

Step 5.4.3.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.3.1

Divide $6$ by $2$ .

$x = 3$

Step 5.5

Solve $x - 2 = - (- x + 4)$ for $x$ .

Tap for more steps...

Step 5.5.1

Simplify $- (- x + 4)$ .

Tap for more steps...

Step 5.5.1.1

Rewrite.

$x - 2 = 0 + 0 - (- x + 4)$

Step 5.5.1.2

Simplify by adding zeros.

$x - 2 = - (- x + 4)$

Step 5.5.1.3

Apply the distributive property.

$x - 2 = - - x - 1 \cdot 4$

Step 5.5.1.4

Multiply $- - x$ .

Tap for more steps...

Step 5.5.1.4.1

Multiply $- 1$ by $- 1$ .

$x - 2 = 1 x - 1 \cdot 4$

Step 5.5.1.4.2

Multiply $x$ by $1$ .

$x - 2 = x - 1 \cdot 4$

Step 5.5.1.5

Multiply $- 1$ by $4$ .

$x - 2 = x - 4$

Step 5.5.2

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

Tap for more steps...

Step 5.5.2.1

Subtract $x$ from both sides of the equation.

$x - 2 - x = - 4$

Step 5.5.2.2

Combine the opposite terms in $x - 2 - x$ .

Tap for more steps...

Step 5.5.2.2.1

Subtract $x$ from $x$ .

$0 - 2 = - 4$

Step 5.5.2.2.2

Subtract $2$ from $0$ .

$- 2 = - 4$

Step 5.5.3

Since $- 2 \neq - 4$ , there are no solutions.

No solution

Step 5.6

Consolidate the solutions.

$x = 3$

Step 6

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

Tap for more steps...

Step 6.1

Solve for $| x - 2 |$ .

Tap for more steps...

Step 6.1.1

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

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

Step 6.1.2

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

Tap for more steps...

Step 6.1.2.1

Apply the distributive property.

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

Step 6.1.2.2

Multiply $- - | x - 2 |$ .

Tap for more steps...

Step 6.1.2.2.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.2.2.2

Multiply $| x - 2 |$ by $1$ .

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

Step 6.1.2.3

Multiply $- 1$ by $3$ .

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

Step 6.1.3

Move all terms not containing $| x - 2 |$ to the right side of the equation.

Tap for more steps...

Step 6.1.3.1

Add $3$ to both sides of the equation.

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

Step 6.1.3.2

Add $- 1$ and $3$ .

$| x - 2 | = x + 2$

Step 6.2

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

$x - 2 = \pm (x + 2)$

Step 6.3

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

$x - 2 = x + 2$

$x - 2 = - (x + 2)$

Step 6.4

Solve $x - 2 = x + 2$ for $x$ .

Tap for more steps...

Step 6.4.1

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

Tap for more steps...

Step 6.4.1.1

Subtract $x$ from both sides of the equation.

$x - 2 - x = 2$

Step 6.4.1.2

Combine the opposite terms in $x - 2 - x$ .

Tap for more steps...

Step 6.4.1.2.1

Subtract $x$ from $x$ .

$0 - 2 = 2$

Step 6.4.1.2.2

Subtract $2$ from $0$ .

$- 2 = 2$

Step 6.4.2

Since $- 2 \neq 2$ , there are no solutions.

No solution

Step 6.5

Solve $x - 2 = - (x + 2)$ for $x$ .

Tap for more steps...

Step 6.5.1

Simplify $- (x + 2)$ .

Tap for more steps...

Step 6.5.1.1

Rewrite.

$x - 2 = 0 + 0 - (x + 2)$

Step 6.5.1.2

Simplify by adding zeros.

$x - 2 = - (x + 2)$

Step 6.5.1.3

Apply the distributive property.

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

Step 6.5.1.4

Multiply $- 1$ by $2$ .

$x - 2 = - x - 2$

Step 6.5.2

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

Tap for more steps...

Step 6.5.2.1

Add $x$ to both sides of the equation.

$x - 2 + x = - 2$

Step 6.5.2.2

Add $x$ and $x$ .

$2 x - 2 = - 2$

Step 6.5.3

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

Tap for more steps...

Step 6.5.3.1

Add $2$ to both sides of the equation.

$2 x = - 2 + 2$

Step 6.5.3.2

Add $- 2$ and $2$ .

$2 x = 0$

Step 6.5.4

Divide each term in $2 x = 0$ by $2$ and simplify.

Tap for more steps...

Step 6.5.4.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 6.5.4.2

Simplify the left side.

Tap for more steps...

Step 6.5.4.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.5.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 6.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 6.5.4.3

Simplify the right side.

Tap for more steps...

Step 6.5.4.3.1

Divide $0$ by $2$ .

$x = 0$

Step 6.6

Consolidate the solutions.

$x = 0$

Step 7

Consolidate the solutions.

$x = 3, 0$

Step 8