Calculus Examples

$0.11 < \frac{1}{x} < 0.14$

Step 1

Solve $0.11 < \frac{1}{x}$ for $x$ .

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Step 1.1

Multiply both sides by $x$ .

$0.11 x = \frac{1}{x} x$

Step 1.2

Simplify the right side.

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Step 1.2.1

Cancel the common factor of $x$ .

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Step 1.2.1.1

Cancel the common factor.

$0.11 x = \frac{1}{x} x$

Step 1.2.1.2

Rewrite the expression.

$0.11 x = 1$

Step 1.3

Divide each term in $0.11 x = 1$ by $0.11$ and simplify.

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Step 1.3.1

Divide each term in $0.11 x = 1$ by $0.11$ .

$\frac{0.11 x}{0.11} = \frac{1}{0.11}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $0.11$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{0.11 x}{0.11} = \frac{1}{0.11}$

Step 1.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{0.11}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Divide $1$ by $0.11$ .

$x = 9.\overline{09}$

Step 1.4

Find the domain of $0.11 - \frac{1}{x}$ .

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Step 1.4.1

Set the denominator in $\frac{1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 1.4.2

The domain is all values of $x$ that make the expression defined.

$(- \infty, 0) \cup (0, \infty)$

Step 1.5

Use each root to create test intervals.

$x < 0$

$0 < x < 9.\overline{09}$

$x > 9.\overline{09}$

Step 1.6

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 1.6.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 1.6.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 1.6.1.2

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

$0.11 < \frac{1}{- 2}$

Step 1.6.1.3

The left side $0.11$ is not less than the right side $- 0.5$ , which means that the given statement is false.

False

Step 1.6.2

Test a value on the interval $0 < x < 9.\overline{09}$ to see if it makes the inequality true.

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Step 1.6.2.1

Choose a value on the interval $0 < x < 9.\overline{09}$ and see if this value makes the original inequality true.

$x = 5$

Step 1.6.2.2

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

$0.11 < \frac{1}{5}$

Step 1.6.2.3

The left side $0.11$ is less than the right side $0.2$ , which means that the given statement is always true.

True

Step 1.6.3

Test a value on the interval $x > 9.\overline{09}$ to see if it makes the inequality true.

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Step 1.6.3.1

Choose a value on the interval $x > 9.\overline{09}$ and see if this value makes the original inequality true.

$x = 12$

Step 1.6.3.2

Replace $x$ with $12$ in the original inequality.

$0.11 < \frac{1}{12}$

Step 1.6.3.3

The left side $0.11$ is not less than the right side $0.08\overline{3}$ , which means that the given statement is false.

False

Step 1.6.4

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

$x < 0$ False

$0 < x < 9.\overline{09}$ True

$x > 9.\overline{09}$ False

$x < 0$ False

$0 < x < 9.\overline{09}$ True

$x > 9.\overline{09}$ False

Step 1.7

The solution consists of all of the true intervals.

$0 < x < 9.\overline{09}$

Step 2

Solve $\frac{1}{x} < 0.14$ for $x$ .

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Step 2.1

Multiply both sides by $x$ .

$\frac{1}{x} x = 0.14 x$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{1}{x} x = 0.14 x$

Step 2.2.1.2

Rewrite the expression.

$1 = 0.14 x$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Rewrite the equation as $0.14 x = 1$ .

$0.14 x = 1$

Step 2.3.2

Divide each term in $0.14 x = 1$ by $0.14$ and simplify.

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Step 2.3.2.1

Divide each term in $0.14 x = 1$ by $0.14$ .

$\frac{0.14 x}{0.14} = \frac{1}{0.14}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $0.14$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{0.14 x}{0.14} = \frac{1}{0.14}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{0.14}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $1$ by $0.14$ .

$x = 7.\overline{142857}$

Step 2.4

Find the domain of $\frac{1}{x} - 0.14$ .

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Step 2.4.1

Set the denominator in $\frac{1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 2.4.2

The domain is all values of $x$ that make the expression defined.

$(- \infty, 0) \cup (0, \infty)$

Step 2.5

Use each root to create test intervals.

$x < 0$

$0 < x < 7.\overline{142857}$

$x > 7.\overline{142857}$

Step 2.6

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 2.6.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 2.6.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 2.6.1.2

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

$\frac{1}{- 2} < 0.14$

Step 2.6.1.3

The left side $- 0.5$ is less than the right side $0.14$ , which means that the given statement is always true.

True

Step 2.6.2

Test a value on the interval $0 < x < 7.\overline{142857}$ to see if it makes the inequality true.

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Step 2.6.2.1

Choose a value on the interval $0 < x < 7.\overline{142857}$ and see if this value makes the original inequality true.

$x = 4$

Step 2.6.2.2

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

$\frac{1}{4} < 0.14$

Step 2.6.2.3

The left side $0.25$ is not less than the right side $0.14$ , which means that the given statement is false.

False

Step 2.6.3

Test a value on the interval $x > 7.\overline{142857}$ to see if it makes the inequality true.

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Step 2.6.3.1

Choose a value on the interval $x > 7.\overline{142857}$ and see if this value makes the original inequality true.

$x = 10$

Step 2.6.3.2

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

$\frac{1}{10} < 0.14$

Step 2.6.3.3

The left side $0.1$ is less than the right side $0.14$ , which means that the given statement is always true.

True

Step 2.6.4

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

$x < 0$ True

$0 < x < 7.\overline{142857}$ False

$x > 7.\overline{142857}$ True

$x < 0$ True

$0 < x < 7.\overline{142857}$ False

$x > 7.\overline{142857}$ True

Step 2.7

The solution consists of all of the true intervals.

$x < 0$ or $x > 7.\overline{142857}$

Step 3

Find the intersection of $0 < x < 9.\overline{09}$ and $x < 0 or x > 7.\overline{142857}$ .

$7.\overline{142857} < x < 9.\overline{09}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$7.\overline{142857} < x < 9.\overline{09}$

Interval Notation:

$(7.\overline{142857}, 9.\overline{09})$

Step 5