Algebra Examples

$\frac{x + 3}{x} \geq 2$

Step 1

Subtract $2$ from both sides of the inequality.

$\frac{x + 3}{x} - 2 \geq 0$

Step 2

Simplify $\frac{x + 3}{x} - 2$ .

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Step 2.1

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

$\frac{x + 3}{x} - 2 \cdot \frac{x}{x} \geq 0$

Step 2.2

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

$\frac{x + 3}{x} + \frac{- 2 x}{x} \geq 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{x + 3 - 2 x}{x} \geq 0$

Step 2.4

Subtract $2 x$ from $x$ .

$\frac{- x + 3}{x} \geq 0$

Step 2.5

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

$\frac{- (x) + 3}{x} \geq 0$

Step 2.6

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

$\frac{- (x) - 1 (- 3)}{x} \geq 0$

Step 2.7

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

$\frac{- (x - 3)}{x} \geq 0$

Step 2.8

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

$\frac{- 1 (x - 3)}{x} \geq 0$

Step 2.9

Move the negative in front of the fraction.

$- \frac{x - 3}{x} \geq 0$

Step 3

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

$x = 0$

$x - 3 = 0$

Step 4

Add $3$ to both sides of the equation.

$x = 3$

Step 5

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

$x = 0$

$x = 3$

Step 6

Consolidate the solutions.

$x = 0, 3$

Step 7

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

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Step 7.1

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

$x = 0$

Step 7.2

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

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

Step 8

Use each root to create test intervals.

$x < 0$

$0 < x < 3$

$x > 3$

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 $x < 0$ to see if it makes the inequality true.

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Step 9.1.1

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

$x = - 2$

Step 9.1.2

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

$\frac{(- 2) + 3}{- 2} \geq 2$

Step 9.1.3

The left side $- 0.5$ is less than the right side $2$ , which means that the given statement is false.

False

Step 9.2

Test a value on the interval $0 < x < 3$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $0 < x < 3$ and see if this value makes the original inequality true.

$x = 2$

Step 9.2.2

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

$\frac{(2) + 3}{2} \geq 2$

Step 9.2.3

The left side $2.5$ is greater than the right side $2$ , which means that the given statement is always true.

True

Step 9.3

Test a value on the interval $x > 3$ to see if it makes the inequality true.

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Step 9.3.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 9.3.2

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

$\frac{(6) + 3}{6} \geq 2$

Step 9.3.3

The left side $1.5$ is less than the right side $2$ , which means that the given statement is false.

False

Step 9.4

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

$x < 0$ False

$0 < x < 3$ True

$x > 3$ False

$x < 0$ False

$0 < x < 3$ True

$x > 3$ False

Step 10

The solution consists of all of the true intervals.

$0 < x \leq 3$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$0 < x \leq 3$

Interval Notation:

$(0, 3]$

Step 12