Precalculus Examples

$\frac{4}{5} > 2 - \frac{3}{x}$

Step 1

Rewrite so $x$ is on the left side of the inequality.

$2 - \frac{3}{x} < \frac{4}{5}$

Step 2

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

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Step 2.1

Subtract $2$ from both sides of the inequality.

$- \frac{3}{x} < \frac{4}{5} - 2$

Step 2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{3}{x} < \frac{4}{5} - 2 \cdot \frac{5}{5}$

Step 2.3

Combine $- 2$ and $\frac{5}{5}$ .

$- \frac{3}{x} < \frac{4}{5} + \frac{- 2 \cdot 5}{5}$

Step 2.4

Combine the numerators over the common denominator.

$- \frac{3}{x} < \frac{4 - 2 \cdot 5}{5}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 2$ by $5$ .

$- \frac{3}{x} < \frac{4 - 10}{5}$

Step 2.5.2

Subtract $10$ from $4$ .

$- \frac{3}{x} < \frac{- 6}{5}$

Step 2.6

Move the negative in front of the fraction.

$- \frac{3}{x} < - \frac{6}{5}$

Step 3

Multiply both sides by $x$ .

$- \frac{3}{x} x = - \frac{6}{5} x$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $x$ .

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Step 4.1.1.1

Move the leading negative in $- \frac{3}{x}$ into the numerator.

$\frac{- 3}{x} x = - \frac{6}{5} x$

Step 4.1.1.2

Cancel the common factor.

$\frac{- 3}{x} x = - \frac{6}{5} x$

Step 4.1.1.3

Rewrite the expression.

$- 3 = - \frac{6}{5} x$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{6}{5} x$ .

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Step 4.2.1.1

Combine $x$ and $\frac{6}{5}$ .

$- 3 = - \frac{x \cdot 6}{5}$

Step 4.2.1.2

Move $6$ to the left of $x$ .

$- 3 = - \frac{6 x}{5}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $- \frac{6 x}{5} = - 3$ .

$- \frac{6 x}{5} = - 3$

Step 5.2

Multiply both sides of the equation by $- \frac{5}{6}$ .

$- \frac{5}{6} (- \frac{6 x}{5}) = - \frac{5}{6} \cdot - 3$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $- \frac{5}{6} (- \frac{6 x}{5})$ .

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Step 5.3.1.1.1

Cancel the common factor of $5$ .

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Step 5.3.1.1.1.1

Move the leading negative in $- \frac{5}{6}$ into the numerator.

$\frac{- 5}{6} (- \frac{6 x}{5}) = - \frac{5}{6} \cdot - 3$

Step 5.3.1.1.1.2

Move the leading negative in $- \frac{6 x}{5}$ into the numerator.

$\frac{- 5}{6} \cdot \frac{- 6 x}{5} = - \frac{5}{6} \cdot - 3$

Step 5.3.1.1.1.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{6} \cdot \frac{- 6 x}{5} = - \frac{5}{6} \cdot - 3$

Step 5.3.1.1.1.4

Cancel the common factor.

$\frac{5 \cdot - 1}{6} \cdot \frac{- 6 x}{5} = - \frac{5}{6} \cdot - 3$

Step 5.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{6} (- 6 x) = - \frac{5}{6} \cdot - 3$

Step 5.3.1.1.2

Cancel the common factor of $6$ .

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Step 5.3.1.1.2.1

Factor $6$ out of $- 6 x$ .

$\frac{- 1}{6} (6 (- x)) = - \frac{5}{6} \cdot - 3$

Step 5.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{6} (6 (- x)) = - \frac{5}{6} \cdot - 3$

Step 5.3.1.1.2.3

Rewrite the expression.

$- - x = - \frac{5}{6} \cdot - 3$

Step 5.3.1.1.3

Multiply.

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Step 5.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{5}{6} \cdot - 3$

Step 5.3.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{5}{6} \cdot - 3$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $- \frac{5}{6} \cdot - 3$ .

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Step 5.3.2.1.1

Cancel the common factor of $3$ .

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Step 5.3.2.1.1.1

Move the leading negative in $- \frac{5}{6}$ into the numerator.

$x = \frac{- 5}{6} \cdot - 3$

Step 5.3.2.1.1.2

Factor $3$ out of $6$ .

$x = \frac{- 5}{3 (2)} \cdot - 3$

Step 5.3.2.1.1.3

Factor $3$ out of $- 3$ .

$x = \frac{- 5}{3 \cdot 2} \cdot (3 \cdot - 1)$

Step 5.3.2.1.1.4

Cancel the common factor.

$x = \frac{- 5}{3 \cdot 2} \cdot (3 \cdot - 1)$

Step 5.3.2.1.1.5

Rewrite the expression.

$x = \frac{- 5}{2} \cdot - 1$

Step 5.3.2.1.2

Combine $\frac{- 5}{2}$ and $- 1$ .

$x = \frac{- 5 \cdot - 1}{2}$

Step 5.3.2.1.3

Multiply $- 5$ by $- 1$ .

$x = \frac{5}{2}$

Step 6

Find the domain of $- \frac{6}{5} + \frac{3}{x}$ .

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Step 6.1

Set the denominator in $\frac{3}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 6.2

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

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

Step 7

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{5}{2}$

$x > \frac{5}{2}$

Step 8

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 8.1

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

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Step 8.1.1

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

$x = - 2$

Step 8.1.2

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

$\frac{4}{5} > 2 - \frac{3}{- 2}$

Step 8.1.3

The left side $0.8$ is not greater than the right side $3.5$ , which means that the given statement is false.

False

Step 8.2

Test a value on the interval $0 < x < \frac{5}{2}$ to see if it makes the inequality true.

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Step 8.2.1

Choose a value on the interval $0 < x < \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 1$

Step 8.2.2

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

$\frac{4}{5} > 2 - \frac{3}{1}$

Step 8.2.3

The left side $0.8$ is greater than the right side $- 1$ , which means that the given statement is always true.

True

Step 8.3

Test a value on the interval $x > \frac{5}{2}$ to see if it makes the inequality true.

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Step 8.3.1

Choose a value on the interval $x > \frac{5}{2}$ and see if this value makes the original inequality true.

$x = 5$

Step 8.3.2

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

$\frac{4}{5} > 2 - \frac{3}{5}$

Step 8.3.3

The left side $0.8$ is not greater than the right side $1.4$ , which means that the given statement is false.

False

Step 8.4

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

$x < 0$ False

$0 < x < \frac{5}{2}$ True

$x > \frac{5}{2}$ False

$x < 0$ False

$0 < x < \frac{5}{2}$ True

$x > \frac{5}{2}$ False

Step 9

The solution consists of all of the true intervals.

$0 < x < \frac{5}{2}$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$0 < x < \frac{5}{2}$

Interval Notation:

$(0, \frac{5}{2})$

Step 11