Solve the Inequality for a (a-1)/a<0

$\frac{a - 1}{a} < 0$

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

$a = 0$

$a - 1 = 0$

Add $1$ to both sides of the equation.

$a = 1$

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

$a = 0$

$a = 1$

Consolidate the solutions.

$a = 0, 1$

Find the domain of $\frac{a - 1}{a}$ .

Set the denominator in $\frac{a - 1}{a}$ equal to $0$ to find where the expression is undefined.

$a = 0$

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

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

Use each root to create test intervals.

$a < 0$

$0 < a < 1$

$a > 1$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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

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

$a = - 2$

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

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

The left side $1.5$ is not less than the right side $0$ , which means that the given statement is false.

False

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

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

$a = 0.5$

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

$\frac{(0.5) - 1}{0.5} < 0$

The left side $- 1$ is less than the right side $0$ , which means that the given statement is always true.

True

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

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

$a = 4$

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

$\frac{(4) - 1}{4} < 0$

The left side $0.75$ is not less than the right side $0$ , which means that the given statement is false.

False

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

$a < 0$ False

$0 < a < 1$ True

$a > 1$ False

$a < 0$ False

$0 < a < 1$ True

$a > 1$ False

The solution consists of all of the true intervals.

$0 < a < 1$

The result can be shown in multiple forms.

Inequality Form:

$0 < a < 1$

Interval Notation:

$(0, 1)$

Solve the Inequality for a (a-1)/a<0

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