Finite Math Examples

$1 + \frac{2}{x} < 3$

Step 1

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

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Step 1.1

Subtract $1$ from both sides of the inequality.

$\frac{2}{x} < 3 - 1$

Step 1.2

Subtract $1$ from $3$ .

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

Step 2

Multiply both sides by $x$ .

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

Step 3

Simplify the left side.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

$2 = 2 x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $2 x = 2$ .

$2 x = 2$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $2$ by $2$ .

$x = 1$

Step 5

Find the domain of $\frac{2}{x} - 2$ .

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Step 5.1

Set the denominator in $\frac{2}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 5.2

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

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

Step 6

Use each root to create test intervals.

$x < 0$

$0 < x < 1$

$x > 1$

Step 7

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 7.1

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

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Step 7.1.1

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

$x = - 2$

Step 7.1.2

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

$1 + \frac{2}{- 2} < 3$

Step 7.1.3

The left side $0$ is less than the right side $3$ , which means that the given statement is always true.

True

Step 7.2

Test a value on the interval $0 < x < 1$ to see if it makes the inequality true.

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Step 7.2.1

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

$x = 0.5$

Step 7.2.2

Replace $x$ with $0.5$ in the original inequality.

$1 + \frac{2}{0.5} < 3$

Step 7.2.3

The left side $5$ is not less than the right side $3$ , which means that the given statement is false.

False

Step 7.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 7.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 7.3.2

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

$1 + \frac{2}{4} < 3$

Step 7.3.3

The left side $1.5$ is less than the right side $3$ , which means that the given statement is always true.

True

Step 7.4

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

$x < 0$ True

$0 < x < 1$ False

$x > 1$ True

$x < 0$ True

$0 < x < 1$ False

$x > 1$ True

Step 8

The solution consists of all of the true intervals.

$x < 0$ or $x > 1$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x < 0 or x > 1$

Interval Notation:

$(- \infty, 0) \cup (1, \infty)$

Step 10