Calculus Examples

$| x - 3 | + | x + 2 | < 11$

Step 1

Replace $<$ with $=$ in $| x - 3 | + | x + 2 | < 11$ .

$| x - 3 | + | x + 2 | = 11$

Step 2

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

$| x + 2 | = 11 - | x - 3 |$

Step 3

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

$x + 2 = \pm (11 - | x - 3 |)$

Step 4

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

$x + 2 = 11 - | x - 3 |$

$x + 2 = - (11 - | x - 3 |)$

Step 5

Solve $x + 2 = 11 - | x - 3 |$ 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 $11 - | x - 3 | = x + 2$ .

$11 - | x - 3 | = x + 2$

Step 5.1.2

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

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Step 5.1.2.1

Subtract $11$ from both sides of the equation.

$- | x - 3 | = x + 2 - 11$

Step 5.1.2.2

Subtract $11$ from $2$ .

$- | x - 3 | = x - 9$

Step 5.1.3

Divide each term in $- | x - 3 | = x - 9$ by $- 1$ and simplify.

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Step 5.1.3.1

Divide each term in $- | x - 3 | = x - 9$ by $- 1$ .

$\frac{- | x - 3 |}{- 1} = \frac{x}{- 1} + \frac{- 9}{- 1}$

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.

$\frac{| x - 3 |}{1} = \frac{x}{- 1} + \frac{- 9}{- 1}$

Step 5.1.3.2.2

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

$| x - 3 | = \frac{x}{- 1} + \frac{- 9}{- 1}$

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

$| x - 3 | = - 1 \cdot x + \frac{- 9}{- 1}$

Step 5.1.3.3.1.2

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

$| x - 3 | = - x + \frac{- 9}{- 1}$

Step 5.1.3.3.1.3

Divide $- 9$ by $- 1$ .

$| x - 3 | = - x + 9$

Step 5.2

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

$x - 3 = \pm (- x + 9)$

Step 5.3

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

$x - 3 = - x + 9$

$x - 3 = - (- x + 9)$

Step 5.4

Solve $x - 3 = - x + 9$ 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.

$x - 3 + x = 9$

Step 5.4.1.2

Add $x$ and $x$ .

$2 x - 3 = 9$

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.

$2 x = 9 + 3$

Step 5.4.2.2

Add $9$ and $3$ .

$2 x = 12$

Step 5.4.3

Divide each term in $2 x = 12$ by $2$ and simplify.

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Step 5.4.3.1

Divide each term in $2 x = 12$ by $2$ .

$\frac{2 x}{2} = \frac{12}{2}$

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.

$\frac{2 x}{2} = \frac{12}{2}$

Step 5.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{2}$

Step 5.4.3.3

Simplify the right side.

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Step 5.4.3.3.1

Divide $12$ by $2$ .

$x = 6$

Step 5.5

Solve $x - 3 = - (- x + 9)$ for $x$ .

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Step 5.5.1

Simplify $- (- x + 9)$ .

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Step 5.5.1.1

Rewrite.

$x - 3 = 0 + 0 - (- x + 9)$

Step 5.5.1.2

Simplify by adding zeros.

$x - 3 = - (- x + 9)$

Step 5.5.1.3

Apply the distributive property.

$x - 3 = - - x - 1 \cdot 9$

Step 5.5.1.4

Multiply $- - x$ .

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Step 5.5.1.4.1

Multiply $- 1$ by $- 1$ .

$x - 3 = 1 x - 1 \cdot 9$

Step 5.5.1.4.2

Multiply $x$ by $1$ .

$x - 3 = x - 1 \cdot 9$

Step 5.5.1.5

Multiply $- 1$ by $9$ .

$x - 3 = x - 9$

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.

$x - 3 - x = - 9$

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 $x$ .

$0 - 3 = - 9$

Step 5.5.2.2.2

Subtract $3$ from $0$ .

$- 3 = - 9$

Step 5.5.3

Since $- 3 \neq - 9$ , there are no solutions.

No solution

Step 5.6

Consolidate the solutions.

$x = 6$

Step 6

Solve $x + 2 = - (11 - | x - 3 |)$ 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 $- (11 - | x - 3 |) = x + 2$ .

$- (11 - | x - 3 |) = x + 2$

Step 6.1.2

Simplify $- (11 - | x - 3 |)$ .

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Step 6.1.2.1

Apply the distributive property.

$- 1 \cdot 11 - - | x - 3 | = x + 2$

Step 6.1.2.2

Multiply $- 1$ by $11$ .

$- 11 - - | x - 3 | = x + 2$

Step 6.1.2.3

Multiply $- - | x - 3 |$ .

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Step 6.1.2.3.1

Multiply $- 1$ by $- 1$ .

$- 11 + 1 | x - 3 | = x + 2$

Step 6.1.2.3.2

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

$- 11 + | x - 3 | = x + 2$

Step 6.1.3

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

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Step 6.1.3.1

Add $11$ to both sides of the equation.

$| x - 3 | = x + 2 + 11$

Step 6.1.3.2

Add $2$ and $11$ .

$| x - 3 | = x + 13$

Step 6.2

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

$x - 3 = \pm (x + 13)$

Step 6.3

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

$x - 3 = x + 13$

$x - 3 = - (x + 13)$

Step 6.4

Solve $x - 3 = x + 13$ 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.

$x - 3 - x = 13$

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 $x$ .

$0 - 3 = 13$

Step 6.4.1.2.2

Subtract $3$ from $0$ .

$- 3 = 13$

Step 6.4.2

Since $- 3 \neq 13$ , there are no solutions.

No solution

Step 6.5

Solve $x - 3 = - (x + 13)$ for $x$ .

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Step 6.5.1

Simplify $- (x + 13)$ .

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Step 6.5.1.1

Rewrite.

$x - 3 = 0 + 0 - (x + 13)$

Step 6.5.1.2

Simplify by adding zeros.

$x - 3 = - (x + 13)$

Step 6.5.1.3

Apply the distributive property.

$x - 3 = - x - 1 \cdot 13$

Step 6.5.1.4

Multiply $- 1$ by $13$ .

$x - 3 = - x - 13$

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.

$x - 3 + x = - 13$

Step 6.5.2.2

Add $x$ and $x$ .

$2 x - 3 = - 13$

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.

$2 x = - 13 + 3$

Step 6.5.3.2

Add $- 13$ and $3$ .

$2 x = - 10$

Step 6.5.4

Divide each term in $2 x = - 10$ by $2$ and simplify.

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Step 6.5.4.1

Divide each term in $2 x = - 10$ by $2$ .

$\frac{2 x}{2} = \frac{- 10}{2}$

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.

$\frac{2 x}{2} = \frac{- 10}{2}$

Step 6.5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 10}{2}$

Step 6.5.4.3

Simplify the right side.

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Step 6.5.4.3.1

Divide $- 10$ by $2$ .

$x = - 5$

Step 6.6

Consolidate the solutions.

$x = - 5$

Step 7

Consolidate the solutions.

$x = 6, - 5$

Step 8

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 6$

$x > 6$

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 $x < - 5$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < - 5$ and see if this value makes the original inequality true.

$x = - 8$

Step 9.1.2

Replace $x$ with $- 8$ in the original inequality.

$| (- 8) - 3 | + | (- 8) + 2 | < 11$

Step 9.1.3

The left side $17$ is not less than the right side $11$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $- 5 < x < 6$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $- 5 < x < 6$ and see if this value makes the original inequality true.

$x = 0$

Step 9.2.2

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

$| (0) - 3 | + | (0) + 2 | < 11$

Step 9.2.3

The left side $5$ is less than the right side $11$ , which means that the given statement is always true.

True

Step 9.3

Test a value on the interval $x > 6$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

Step 9.3.2

Replace $x$ with $8$ in the original inequality.

$| (8) - 3 | + | (8) + 2 | < 11$

Step 9.3.3

The left side $15$ is not less than the right side $11$ , which means that the given statement is false.

False

Step 9.4

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

$x < - 5$ False

$- 5 < x < 6$ True

$x > 6$ False

$x < - 5$ False

$- 5 < x < 6$ True

$x > 6$ False

Step 10

The solution consists of all of the true intervals.

$- 5 < x < 6$

Step 11

Convert the inequality to interval notation.

$(- 5, 6)$

Step 12