Algebra Examples

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

Step 1

Subtract $5$ from both sides of the inequality.

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

Step 2

Simplify $\frac{2 x}{x - 2} - 5$ .

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Step 2.1

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

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

Step 2.2

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

$\frac{2 x}{x - 2} + \frac{- 5 (x - 2)}{x - 2} < 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{2 x - 5 (x - 2)}{x - 2} < 0$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $- 5$ by $- 2$ .

$\frac{2 x - 5 x + 10}{x - 2} < 0$

Step 2.4.3

Subtract $5 x$ from $2 x$ .

$\frac{- 3 x + 10}{x - 2} < 0$

Step 2.5

Factor $- 1$ out of $- 3 x$ .

$\frac{- (3 x) + 10}{x - 2} < 0$

Step 2.6

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{- (3 x) - 1 (- 10)}{x - 2} < 0$

Step 2.7

Factor $- 1$ out of $- (3 x) - 1 (- 10)$ .

$\frac{- (3 x - 10)}{x - 2} < 0$

Step 2.8

Rewrite $- (3 x - 10)$ as $- 1 (3 x - 10)$ .

$\frac{- 1 (3 x - 10)}{x - 2} < 0$

Step 2.9

Move the negative in front of the fraction.

$- \frac{3 x - 10}{x - 2} < 0$

Step 3

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$3 x - 10 = 0$

$x - 2 = 0$

Step 4

Add $10$ to both sides of the equation.

$3 x = 10$

Step 5

Divide each term in $3 x = 10$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = 10$ by $3$ .

$\frac{3 x}{3} = \frac{10}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{10}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{3}$

Step 6

Add $2$ to both sides of the equation.

$x = 2$

Step 7

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = \frac{10}{3}$

$x = 2$

Step 8

Consolidate the solutions.

$x = \frac{10}{3}, 2$

Step 9

Find the domain of $- \frac{3 x - 10}{x - 2}$ .

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Step 9.1

Set the denominator in $\frac{3 x - 10}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Step 9.2

Add $2$ to both sides of the equation.

$x = 2$

Step 9.3

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

$(- \infty, 2) \cup (2, \infty)$

Step 10

Use each root to create test intervals.

$x < 2$

$2 < x < \frac{10}{3}$

$x > \frac{10}{3}$

Step 11

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 11.1

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

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Step 11.1.1

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

$x = 0$

Step 11.1.2

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

$\frac{2 (0)}{(0) - 2} < 5$

Step 11.1.3

The left side $0$ is less than the right side $5$ , which means that the given statement is always true.

True

Step 11.2

Test a value on the interval $2 < x < \frac{10}{3}$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $2 < x < \frac{10}{3}$ and see if this value makes the original inequality true.

$x = 3$

Step 11.2.2

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

$\frac{2 (3)}{(3) - 2} < 5$

Step 11.2.3

The left side $6$ is not less than the right side $5$ , which means that the given statement is false.

False

Step 11.3

Test a value on the interval $x > \frac{10}{3}$ to see if it makes the inequality true.

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Step 11.3.1

Choose a value on the interval $x > \frac{10}{3}$ and see if this value makes the original inequality true.

$x = 6$

Step 11.3.2

Replace $x$ with $6$ in the original inequality.

$\frac{2 (6)}{(6) - 2} < 5$

Step 11.3.3

The left side $3$ is less than the right side $5$ , which means that the given statement is always true.

True

Step 11.4

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

$x < 2$ True

$2 < x < \frac{10}{3}$ False

$x > \frac{10}{3}$ True

$x < 2$ True

$2 < x < \frac{10}{3}$ False

$x > \frac{10}{3}$ True

Step 12

The solution consists of all of the true intervals.

$x < 2$ or $x > \frac{10}{3}$

Step 13

The result can be shown in multiple forms.

Inequality Form:

$x < 2 or x > \frac{10}{3}$

Interval Notation:

$(- \infty, 2) \cup (\frac{10}{3}, \infty)$

Step 14