Pre-Algebra Examples

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

Step 1

Replace $>$ with $=$ in $| x + 1 | + | 3 x - 1 | > 2$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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

Step 5

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

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Step 5.1

Solve for $| x + 1 |$ .

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Step 5.1.1

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

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

Step 5.1.2

Move all terms not containing $| x + 1 |$ to the right side of the equation.

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Step 5.1.2.1

Subtract $2$ from both sides of the equation.

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

Step 5.1.2.2

Subtract $2$ from $- 1$ .

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

Step 5.1.3

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

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Step 5.1.3.1

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

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

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.

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

Step 5.1.3.2.2

Divide $| x + 1 |$ by $1$ .

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

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 $\frac{3 x}{- 1}$ .

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

Step 5.1.3.3.1.2

Rewrite $- 1 \cdot (3 x)$ as $- (3 x)$ .

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

Step 5.1.3.3.1.3

Multiply $3$ by $- 1$ .

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

Step 5.1.3.3.1.4

Divide $- 3$ by $- 1$ .

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

Step 5.2

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

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

Step 5.3

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

$x + 1 = - 3 x + 3$

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

Step 5.4

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

Add $3 x$ to both sides of the equation.

$x + 1 + 3 x = 3$

Step 5.4.1.2

Add $x$ and $3 x$ .

$4 x + 1 = 3$

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

Subtract $1$ from both sides of the equation.

$4 x = 3 - 1$

Step 5.4.2.2

Subtract $1$ from $3$ .

$4 x = 2$

Step 5.4.3

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

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Step 5.4.3.1

Divide each term in $4 x = 2$ by $4$ .

$\frac{4 x}{4} = \frac{2}{4}$

Step 5.4.3.2

Simplify the left side.

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Step 5.4.3.2.1

Cancel the common factor of $4$ .

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Step 5.4.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{2}{4}$

Step 5.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{4}$

Step 5.4.3.3

Simplify the right side.

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Step 5.4.3.3.1

Cancel the common factor of $2$ and $4$ .

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Step 5.4.3.3.1.1

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{4}$

Step 5.4.3.3.1.2

Cancel the common factors.

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Step 5.4.3.3.1.2.1

Factor $2$ out of $4$ .

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

Step 5.4.3.3.1.2.2

Cancel the common factor.

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

Step 5.4.3.3.1.2.3

Rewrite the expression.

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

Step 5.5

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

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Step 5.5.1

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

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Step 5.5.1.1

Rewrite.

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

Step 5.5.1.2

Simplify by adding zeros.

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

Step 5.5.1.3

Apply the distributive property.

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

Step 5.5.1.4

Multiply.

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Step 5.5.1.4.1

Multiply $- 3$ by $- 1$ .

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

Step 5.5.1.4.2

Multiply $- 1$ by $3$ .

$x + 1 = 3 x - 3$

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

Subtract $3 x$ from both sides of the equation.

$x + 1 - 3 x = - 3$

Step 5.5.2.2

Subtract $3 x$ from $x$ .

$- 2 x + 1 = - 3$

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

Subtract $1$ from both sides of the equation.

$- 2 x = - 3 - 1$

Step 5.5.3.2

Subtract $1$ from $- 3$ .

$- 2 x = - 4$

Step 5.5.4

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

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Step 5.5.4.1

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

$\frac{- 2 x}{- 2} = \frac{- 4}{- 2}$

Step 5.5.4.2

Simplify the left side.

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Step 5.5.4.2.1

Cancel the common factor of $- 2$ .

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Step 5.5.4.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 4}{- 2}$

Step 5.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 2}$

Step 5.5.4.3

Simplify the right side.

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Step 5.5.4.3.1

Divide $- 4$ by $- 2$ .

$x = 2$

Step 5.6

Consolidate the solutions.

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

Step 6

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

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Step 6.1

Solve for $| x + 1 |$ .

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Step 6.1.1

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

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

Step 6.1.2

Simplify $- (2 - | x + 1 |)$ .

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Step 6.1.2.1

Apply the distributive property.

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

Step 6.1.2.2

Multiply $- 1$ by $2$ .

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

Step 6.1.2.3

Multiply $- - | x + 1 |$ .

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Step 6.1.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.1.2.3.2

Multiply $| x + 1 |$ by $1$ .

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

Step 6.1.3

Move all terms not containing $| x + 1 |$ to the right side of the equation.

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Step 6.1.3.1

Add $2$ to both sides of the equation.

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

Step 6.1.3.2

Add $- 1$ and $2$ .

$| x + 1 | = 3 x + 1$

Step 6.2

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

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

Step 6.3

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

$x + 1 = 3 x + 1$

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

Step 6.4

Solve $x + 1 = 3 x + 1$ 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 $3 x$ from both sides of the equation.

$x + 1 - 3 x = 1$

Step 6.4.1.2

Subtract $3 x$ from $x$ .

$- 2 x + 1 = 1$

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

Subtract $1$ from both sides of the equation.

$- 2 x = 1 - 1$

Step 6.4.2.2

Subtract $1$ from $1$ .

$- 2 x = 0$

Step 6.4.3

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

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Step 6.4.3.1

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

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

Step 6.4.3.2

Simplify the left side.

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Step 6.4.3.2.1

Cancel the common factor of $- 2$ .

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Step 6.4.3.2.1.1

Cancel the common factor.

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

Step 6.4.3.2.1.2

Divide $x$ by $1$ .

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

Step 6.4.3.3

Simplify the right side.

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Step 6.4.3.3.1

Divide $0$ by $- 2$ .

$x = 0$

Step 6.5

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

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Step 6.5.1

Simplify $- (3 x + 1)$ .

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Step 6.5.1.1

Rewrite.

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

Step 6.5.1.2

Simplify by adding zeros.

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

Step 6.5.1.3

Apply the distributive property.

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

Step 6.5.1.4

Multiply.

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Step 6.5.1.4.1

Multiply $3$ by $- 1$ .

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

Step 6.5.1.4.2

Multiply $- 1$ by $1$ .

$x + 1 = - 3 x - 1$

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 $3 x$ to both sides of the equation.

$x + 1 + 3 x = - 1$

Step 6.5.2.2

Add $x$ and $3 x$ .

$4 x + 1 = - 1$

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

Subtract $1$ from both sides of the equation.

$4 x = - 1 - 1$

Step 6.5.3.2

Subtract $1$ from $- 1$ .

$4 x = - 2$

Step 6.5.4

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

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Step 6.5.4.1

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

$\frac{4 x}{4} = \frac{- 2}{4}$

Step 6.5.4.2

Simplify the left side.

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Step 6.5.4.2.1

Cancel the common factor of $4$ .

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Step 6.5.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 2}{4}$

Step 6.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{4}$

Step 6.5.4.3

Simplify the right side.

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Step 6.5.4.3.1

Cancel the common factor of $- 2$ and $4$ .

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Step 6.5.4.3.1.1

Factor $2$ out of $- 2$ .

$x = \frac{2 (- 1)}{4}$

Step 6.5.4.3.1.2

Cancel the common factors.

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Step 6.5.4.3.1.2.1

Factor $2$ out of $4$ .

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

Step 6.5.4.3.1.2.2

Cancel the common factor.

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

Step 6.5.4.3.1.2.3

Rewrite the expression.

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

Step 6.5.4.3.2

Move the negative in front of the fraction.

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

Step 6.6

Consolidate the solutions.

$x = 0, - \frac{1}{2}$

Step 7

Consolidate the solutions.

$x = \frac{1}{2}, 2, 0, - \frac{1}{2}$

Step 8

Use each root to create test intervals.

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

$- \frac{1}{2} < x < 0$

$0 < x < \frac{1}{2}$

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

$x > 2$

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.

$x = - 3$

Step 9.1.2

Replace $x$ with $- 3$ in the original inequality.

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

Step 9.1.3

The left side $12$ is greater than the right side $2$ , which means that the given statement is always true.

True

Step 9.2

Test a value on the interval $- \frac{1}{2} < x < 0$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- \frac{1}{2} < x < 0$ and see if this value makes the original inequality true.

$x = - 0.25$

Step 9.2.2

Replace $x$ with $- 0.25$ in the original inequality.

$| (- 0.25) + 1 | + | 3 (- 0.25) - 1 | > 2$

Step 9.2.3

The left side $2.5$ is greater than the right side $2$ , which means that the given statement is always true.

True

Step 9.3

Test a value on the interval $0 < x < \frac{1}{2}$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $0 < x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0.25$

Step 9.3.2

Replace $x$ with $0.25$ in the original inequality.

$| (0.25) + 1 | + | 3 (0.25) - 1 | > 2$

Step 9.3.3

The left side $1.5$ is not greater than the right side $2$ , which means that the given statement is false.

False

Step 9.4

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

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Step 9.4.1

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

$x = 1$

Step 9.4.2

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

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

Step 9.4.3

The left side $4$ is greater than the right side $2$ , which means that the given statement is always true.

True

Step 9.5

Test a value on the interval $x > 2$ to see if it makes the inequality true.

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Step 9.5.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 9.5.2

Replace $x$ with $4$ in the original inequality.

$| (4) + 1 | + | 3 (4) - 1 | > 2$

Step 9.5.3

The left side $16$ is greater than the right side $2$ , which means that the given statement is always true.

True

Step 9.6

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

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

$- \frac{1}{2} < x < 0$ True

$0 < x < \frac{1}{2}$ False

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

$x > 2$ True

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

$- \frac{1}{2} < x < 0$ True

$0 < x < \frac{1}{2}$ False

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

$x > 2$ True

Step 10

The solution consists of all of the true intervals.

$x < - \frac{1}{2}$ or $- \frac{1}{2} < x < 0$ or $\frac{1}{2} < x < 2$ or $x > 2$

Step 11