Algebra Examples

$\frac{x - 10}{x} < 1$

Step 1

Solve $x > 0$ .

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Step 1.1

Subtract $1$ from both sides of the inequality.

$\frac{x - 10}{x} - 1 < 0$

Step 1.2

Simplify $\frac{x - 10}{x} - 1$ .

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Step 1.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x - 10}{x} - 1 \cdot \frac{x}{x} < 0$

Step 1.2.2

Combine $- 1$ and $\frac{x}{x}$ .

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Subtract $x$ from $x$ .

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

Step 1.2.4.2

Subtract $10$ from $0$ .

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

Step 1.2.5

Move the negative in front of the fraction.

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

Step 1.3

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

$x = 0$

Step 1.4

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

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Step 1.4.1

Set the denominator in $\frac{10}{x}$ 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

The solution consists of all of the true intervals.

$x > 0$

Step 2

Use the inequality $x > 0$ to build the set notation.

${x | x > 0}$

Step 3