Solve the Inequality for x (x-2)/(2x-3)<=0

$\frac{x - 2}{2 x - 3} \leq 0$

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

$x - 2 = 0$

$2 x - 3 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

Divide each term in $2 x = 3$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

$x = 2$

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

Consolidate the solutions.

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

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

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

$2 x - 3 = 0$

Solve for $x$ .

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

Divide each term in $2 x = 3$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

Use each root to create test intervals.

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

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

$x > 2$

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 $x < \frac{3}{2}$ to see if it makes the inequality true.

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

$x = 0$

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

$\frac{(0) - 2}{2 (0) - 3} \leq 0$

The left side $0.\overline{6}$ is greater than the right side $0$ , which means that the given statement is false.

False

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

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

$x = 1.75$

Replace $x$ with $1.75$ in the original inequality.

$\frac{(1.75) - 2}{2 (1.75) - 3} \leq 0$

The left side $- 0.5$ is less than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = 4$

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

$\frac{(4) - 2}{2 (4) - 3} \leq 0$

The left side $0.4$ is greater 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.

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

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

$x > 2$ False

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

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

$x > 2$ False

The solution consists of all of the true intervals.

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

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(\frac{3}{2}, 2]$

Solve the Inequality for x (x-2)/(2x-3)<=0

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