Algebra Examples

$| 2 x - 4 | - | x | \leq 3$

Step 1

Replace $\leq$ with $=$ in $| 2 x - 4 | - | x | \leq 3$ .

$| 2 x - 4 | - | x | = 3$

Step 2

Solve for $| x |$ .

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Step 2.1

Subtract $| 2 x - 4 |$ from both sides of the equation.

$- | x | = 3 - | 2 x - 4 |$

Step 2.2

Divide each term in $- | x | = 3 - | 2 x - 4 |$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- | x | = 3 - | 2 x - 4 |$ by $- 1$ .

$\frac{- | x |}{- 1} = \frac{3}{- 1} + \frac{- | 2 x - 4 |}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{| x |}{1} = \frac{3}{- 1} + \frac{- | 2 x - 4 |}{- 1}$

Step 2.2.2.2

Divide $| x |$ by $1$ .

$| x | = \frac{3}{- 1} + \frac{- | 2 x - 4 |}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Divide $3$ by $- 1$ .

$| x | = - 3 + \frac{- | 2 x - 4 |}{- 1}$

Step 2.2.3.1.2

Dividing two negative values results in a positive value.

$| x | = - 3 + \frac{| 2 x - 4 |}{1}$

Step 2.2.3.1.3

Divide $| 2 x - 4 |$ by $1$ .

$| x | = - 3 + | 2 x - 4 |$

Step 3

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

$x = \pm (- 3 + | 2 x - 4 |)$

Step 4

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

$x = - 3 + | 2 x - 4 |$

$x = - (- 3 + | 2 x - 4 |)$

Step 5

Solve $x = - 3 + | 2 x - 4 |$ for $x$ .

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Step 5.1

Solve for $| 2 x - 4 |$ .

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Step 5.1.1

Rewrite the equation as $- 3 + | 2 x - 4 | = x$ .

$- 3 + | 2 x - 4 | = x$

Step 5.1.2

Add $3$ to both sides of the equation.

$| 2 x - 4 | = x + 3$

Step 5.2

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

$2 x - 4 = \pm (x + 3)$

Step 5.3

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

$2 x - 4 = x + 3$

$2 x - 4 = - (x + 3)$

Step 5.4

Solve $2 x - 4 = x + 3$ 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

Subtract $x$ from both sides of the equation.

$2 x - 4 - x = 3$

Step 5.4.1.2

Subtract $x$ from $2 x$ .

$x - 4 = 3$

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 $4$ to both sides of the equation.

$x = 3 + 4$

Step 5.4.2.2

Add $3$ and $4$ .

$x = 7$

Step 5.5

Solve $2 x - 4 = - (x + 3)$ for $x$ .

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Step 5.5.1

Simplify $- (x + 3)$ .

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Step 5.5.1.1

Rewrite.

$2 x - 4 = 0 + 0 - (x + 3)$

Step 5.5.1.2

Simplify by adding zeros.

$2 x - 4 = - (x + 3)$

Step 5.5.1.3

Apply the distributive property.

$2 x - 4 = - x - 1 \cdot 3$

Step 5.5.1.4

Multiply $- 1$ by $3$ .

$2 x - 4 = - x - 3$

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

Add $x$ to both sides of the equation.

$2 x - 4 + x = - 3$

Step 5.5.2.2

Add $2 x$ and $x$ .

$3 x - 4 = - 3$

Step 5.5.3

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

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Step 5.5.3.1

Add $4$ to both sides of the equation.

$3 x = - 3 + 4$

Step 5.5.3.2

Add $- 3$ and $4$ .

$3 x = 1$

Step 5.5.4

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

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Step 5.5.4.1

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

$\frac{3 x}{3} = \frac{1}{3}$

Step 5.5.4.2

Simplify the left side.

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Step 5.5.4.2.1

Cancel the common factor of $3$ .

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Step 5.5.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 5.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 5.6

Consolidate the solutions.

$x = 7, \frac{1}{3}$

Step 6

Solve $x = - (- 3 + | 2 x - 4 |)$ for $x$ .

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Step 6.1

Solve for $| 2 x - 4 |$ .

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Step 6.1.1

Rewrite the equation as $- (- 3 + | 2 x - 4 |) = x$ .

$- (- 3 + | 2 x - 4 |) = x$

Step 6.1.2

Simplify $- (- 3 + | 2 x - 4 |)$ .

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Step 6.1.2.1

Apply the distributive property.

$- - 3 - | 2 x - 4 | = x$

Step 6.1.2.2

Multiply $- 1$ by $- 3$ .

$3 - | 2 x - 4 | = x$

Step 6.1.3

Subtract $3$ from both sides of the equation.

$- | 2 x - 4 | = x - 3$

Step 6.1.4

Divide each term in $- | 2 x - 4 | = x - 3$ by $- 1$ and simplify.

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Step 6.1.4.1

Divide each term in $- | 2 x - 4 | = x - 3$ by $- 1$ .

$\frac{- | 2 x - 4 |}{- 1} = \frac{x}{- 1} + \frac{- 3}{- 1}$

Step 6.1.4.2

Simplify the left side.

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Step 6.1.4.2.1

Dividing two negative values results in a positive value.

$\frac{| 2 x - 4 |}{1} = \frac{x}{- 1} + \frac{- 3}{- 1}$

Step 6.1.4.2.2

Divide $| 2 x - 4 |$ by $1$ .

$| 2 x - 4 | = \frac{x}{- 1} + \frac{- 3}{- 1}$

Step 6.1.4.3

Simplify the right side.

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Step 6.1.4.3.1

Simplify each term.

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Step 6.1.4.3.1.1

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

$| 2 x - 4 | = - 1 \cdot x + \frac{- 3}{- 1}$

Step 6.1.4.3.1.2

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

$| 2 x - 4 | = - x + \frac{- 3}{- 1}$

Step 6.1.4.3.1.3

Divide $- 3$ by $- 1$ .

$| 2 x - 4 | = - x + 3$

Step 6.2

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

$2 x - 4 = \pm (- x + 3)$

Step 6.3

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

$2 x - 4 = - x + 3$

$2 x - 4 = - (- x + 3)$

Step 6.4

Solve $2 x - 4 = - x + 3$ 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

Add $x$ to both sides of the equation.

$2 x - 4 + x = 3$

Step 6.4.1.2

Add $2 x$ and $x$ .

$3 x - 4 = 3$

Step 6.4.2

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

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Step 6.4.2.1

Add $4$ to both sides of the equation.

$3 x = 3 + 4$

Step 6.4.2.2

Add $3$ and $4$ .

$3 x = 7$

Step 6.4.3

Divide each term in $3 x = 7$ by $3$ and simplify.

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Step 6.4.3.1

Divide each term in $3 x = 7$ by $3$ .

$\frac{3 x}{3} = \frac{7}{3}$

Step 6.4.3.2

Simplify the left side.

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Step 6.4.3.2.1

Cancel the common factor of $3$ .

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Step 6.4.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{7}{3}$

Step 6.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{3}$

Step 6.5

Solve $2 x - 4 = - (- x + 3)$ for $x$ .

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Step 6.5.1

Simplify $- (- x + 3)$ .

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Step 6.5.1.1

Rewrite.

$2 x - 4 = 0 + 0 - (- x + 3)$

Step 6.5.1.2

Simplify by adding zeros.

$2 x - 4 = - (- x + 3)$

Step 6.5.1.3

Apply the distributive property.

$2 x - 4 = - - x - 1 \cdot 3$

Step 6.5.1.4

Multiply $- - x$ .

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Step 6.5.1.4.1

Multiply $- 1$ by $- 1$ .

$2 x - 4 = 1 x - 1 \cdot 3$

Step 6.5.1.4.2

Multiply $x$ by $1$ .

$2 x - 4 = x - 1 \cdot 3$

Step 6.5.1.5

Multiply $- 1$ by $3$ .

$2 x - 4 = x - 3$

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

Subtract $x$ from both sides of the equation.

$2 x - 4 - x = - 3$

Step 6.5.2.2

Subtract $x$ from $2 x$ .

$x - 4 = - 3$

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 $4$ to both sides of the equation.

$x = - 3 + 4$

Step 6.5.3.2

Add $- 3$ and $4$ .

$x = 1$

Step 6.6

Consolidate the solutions.

$x = \frac{7}{3}, 1$

Step 7

Consolidate the solutions.

$x = 7, \frac{1}{3}, \frac{7}{3}$

Step 8

Use each root to create test intervals.

$x < \frac{1}{3}$

$\frac{1}{3} < x < \frac{7}{3}$

$\frac{7}{3} < x < 7$

$x > 7$

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 $x < \frac{1}{3}$ to see if it makes the inequality true.

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Step 9.1.1

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

$x = 0$

Step 9.1.2

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

$| 2 (0) - 4 | - | 0 | \leq 3$

Step 9.1.3

The left side $4$ is greater than the right side $3$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $\frac{1}{3} < x < \frac{7}{3}$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $\frac{1}{3} < x < \frac{7}{3}$ and see if this value makes the original inequality true.

$x = 1$

Step 9.2.2

Replace $x$ with $1$ in the original inequality.

$| 2 (1) - 4 | - | 1 | \leq 3$

Step 9.2.3

The left side $1$ is less than the right side $3$ , which means that the given statement is always true.

True

Step 9.3

Test a value on the interval $\frac{7}{3} < x < 7$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $\frac{7}{3} < x < 7$ and see if this value makes the original inequality true.

$x = 5$

Step 9.3.2

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

$| 2 (5) - 4 | - | 5 | \leq 3$

Step 9.3.3

The left side $1$ is less than the right side $3$ , which means that the given statement is always true.

True

Step 9.4

Test a value on the interval $x > 7$ to see if it makes the inequality true.

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Step 9.4.1

Choose a value on the interval $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Step 9.4.2

Replace $x$ with $10$ in the original inequality.

$| 2 (10) - 4 | - | 10 | \leq 3$

Step 9.4.3

The left side $6$ is greater than the right side $3$ , which means that the given statement is false.

False

Step 9.5

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

$x < \frac{1}{3}$ False

$\frac{1}{3} < x < \frac{7}{3}$ True

$\frac{7}{3} < x < 7$ True

$x > 7$ False

$x < \frac{1}{3}$ False

$\frac{1}{3} < x < \frac{7}{3}$ True

$\frac{7}{3} < x < 7$ True

$x > 7$ False

Step 10

The solution consists of all of the true intervals.

$\frac{1}{3} \leq x \leq \frac{7}{3}$ or $\frac{7}{3} \leq x \leq 7$

Step 11

Combine the intervals.

$\frac{1}{3} \leq x \leq 7$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$\frac{1}{3} \leq x \leq 7$

Interval Notation:

$[\frac{1}{3}, 7]$

Step 13