Graph (x-1)/(x-3)<0

$\frac{x - 1}{x - 3} < 0$

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

$x - 1 = 0$

$x - 3 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Add $3$ to both sides of the equation.

$x = 3$

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

$x = 1$

$x = 3$

Consolidate the solutions.

$x = 1, 3$

Find the domain of $\frac{x - 1}{x - 3}$ .

Set the denominator in $\frac{x - 1}{x - 3}$ equal to $0$ to find where the expression is undefined.

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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

$(- \infty, 3) \cup (3, \infty)$

Use each root to create test intervals.

$x < 1$

$1 < x < 3$

$x > 3$

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 $x < 1$ to see if it makes the inequality true.

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

$x = 0$

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

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

The left side $0.\overline{3}$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $1 < x < 3$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 3$ and see if this value makes the original inequality true.

$x = 2$

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

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

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 $x > 3$ to see if it makes the inequality true.

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

$x = 6$

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

$\frac{(6) - 1}{(6) - 3} < 0$

The left side $1.\overline{6}$ 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.

$x < 1$ False

$1 < x < 3$ True

$x > 3$ False

$x < 1$ False

$1 < x < 3$ True

$x > 3$ False

The solution consists of all of the true intervals.

$1 < x < 3$

Graph (x-1)/(x-3)<0

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