Algebra Examples

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

Step 1

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

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

Step 2

Solve for $| x |$ .

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Step 2.1

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

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

Step 2.2

Subtract $2$ from both sides of the equation.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.2.2

Divide $| x |$ by $1$ .

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

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

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

Step 2.3.3.1.2

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

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

Step 2.3.3.1.3

Divide $- 2$ by $- 1$ .

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

Step 3

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

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

Step 4

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

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

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

Step 5

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

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Step 5.1

Solve for $| x - 3 |$ .

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Step 5.1.1

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

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

Step 5.1.2

Subtract $2$ from both sides of the equation.

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

Step 5.1.3

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

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Step 5.1.3.1

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

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

Step 5.1.3.2

Simplify the left side.

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Step 5.1.3.2.1

Dividing two negative values results in a positive value.

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

Step 5.1.3.2.2

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

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

Step 5.1.3.3

Simplify the right side.

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Step 5.1.3.3.1

Simplify each term.

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Step 5.1.3.3.1.1

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

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

Step 5.1.3.3.1.2

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

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

Step 5.1.3.3.1.3

Divide $- 2$ by $- 1$ .

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

Step 5.2

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

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

Step 5.3

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

$x - 3 = - x + 2$

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

Step 5.4

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

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Step 5.4.1

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

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Step 5.4.1.1

Add $x$ to both sides of the equation.

$x - 3 + x = 2$

Step 5.4.1.2

Add $x$ and $x$ .

$2 x - 3 = 2$

Step 5.4.2

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

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Step 5.4.2.1

Add $3$ to both sides of the equation.

$2 x = 2 + 3$

Step 5.4.2.2

Add $2$ and $3$ .

$2 x = 5$

Step 5.4.3

Divide each term in $2 x = 5$ by $2$ and simplify.

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Step 5.4.3.1

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Step 5.4.3.2

Simplify the left side.

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Step 5.4.3.2.1

Cancel the common factor of $2$ .

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Step 5.4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Step 5.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 5.5

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

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Step 5.5.1

Simplify $- (- x + 2)$ .

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Step 5.5.1.1

Rewrite.

$x - 3 = 0 + 0 - (- x + 2)$

Step 5.5.1.2

Simplify by adding zeros.

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

Step 5.5.1.3

Apply the distributive property.

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

Step 5.5.1.4

Multiply $- - x$ .

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Step 5.5.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 5.5.1.4.2

Multiply $x$ by $1$ .

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

Step 5.5.1.5

Multiply $- 1$ by $2$ .

$x - 3 = x - 2$

Step 5.5.2

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

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Step 5.5.2.1

Subtract $x$ from both sides of the equation.

$x - 3 - x = - 2$

Step 5.5.2.2

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

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Step 5.5.2.2.1

Subtract $x$ from $x$ .

$0 - 3 = - 2$

Step 5.5.2.2.2

Subtract $3$ from $0$ .

$- 3 = - 2$

Step 5.5.3

Since $- 3 \neq - 2$ , there are no solutions.

No solution

Step 5.6

Consolidate the solutions.

$x = \frac{5}{2}$

Step 6

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

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Step 6.1

Solve for $| x - 3 |$ .

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Step 6.1.1

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

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

Step 6.1.2

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

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Step 6.1.2.1

Apply the distributive property.

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

Step 6.1.2.2

Multiply $- - | x - 3 |$ .

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Step 6.1.2.2.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.2.2.2

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

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

Step 6.1.2.3

Multiply $- 1$ by $2$ .

$| x - 3 | - 2 = x$

Step 6.1.3

Add $2$ to both sides of the equation.

$| x - 3 | = x + 2$

Step 6.2

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

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

Step 6.3

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

$x - 3 = x + 2$

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

Step 6.4

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

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Step 6.4.1

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

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Step 6.4.1.1

Subtract $x$ from both sides of the equation.

$x - 3 - x = 2$

Step 6.4.1.2

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

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Step 6.4.1.2.1

Subtract $x$ from $x$ .

$0 - 3 = 2$

Step 6.4.1.2.2

Subtract $3$ from $0$ .

$- 3 = 2$

Step 6.4.2

Since $- 3 \neq 2$ , there are no solutions.

No solution

Step 6.5

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

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Step 6.5.1

Simplify $- (x + 2)$ .

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Step 6.5.1.1

Rewrite.

$x - 3 = 0 + 0 - (x + 2)$

Step 6.5.1.2

Simplify by adding zeros.

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

Step 6.5.1.3

Apply the distributive property.

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

Step 6.5.1.4

Multiply $- 1$ by $2$ .

$x - 3 = - x - 2$

Step 6.5.2

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

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Step 6.5.2.1

Add $x$ to both sides of the equation.

$x - 3 + x = - 2$

Step 6.5.2.2

Add $x$ and $x$ .

$2 x - 3 = - 2$

Step 6.5.3

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

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Step 6.5.3.1

Add $3$ to both sides of the equation.

$2 x = - 2 + 3$

Step 6.5.3.2

Add $- 2$ and $3$ .

$2 x = 1$

Step 6.5.4

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

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Step 6.5.4.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 6.5.4.2

Simplify the left side.

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Step 6.5.4.2.1

Cancel the common factor of $2$ .

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Step 6.5.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 6.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 6.6

Consolidate the solutions.

$x = \frac{1}{2}$

Step 7

Consolidate the solutions.

$x = \frac{5}{2}, \frac{1}{2}$

Step 8

Use each root to create test intervals.

$x < \frac{1}{2}$

$\frac{1}{2} < x < \frac{5}{2}$

$x > \frac{5}{2}$

Step 9

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

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Step 9.1

Test a value on the interval $x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0$

Step 9.1.2

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

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

Step 9.1.3

The left side $3$ does not equal to the right side $2$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $\frac{1}{2} < x < \frac{5}{2}$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $\frac{1}{2} < x < \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 2$

Step 9.2.2

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

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

Step 9.2.3

The left side $3$ does not equal to the right side $2$ , which means that the given statement is false.

False

Step 9.3

Test a value on the interval $x > \frac{5}{2}$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $x > \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 5$

Step 9.3.2

Replace $x$ with $5$ in the original inequality.

$| 5 | + | (5) - 3 | = 2$

Step 9.3.3

The left side $7$ does not equal to the right side $2$ , which means that the given statement is false.

False

Step 9.4

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

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < \frac{5}{2}$ False

$x > \frac{5}{2}$ False

$x < \frac{1}{2}$ False

$\frac{1}{2} < x < \frac{5}{2}$ False

$x > \frac{5}{2}$ False

Step 10

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution