Convert to Interval Notation (2x-7)/(x-2)>1

$\frac{2 x - 7}{x - 2} > 1$

Step 1

Subtract $1$ from both sides of the inequality.

$\frac{2 x - 7}{x - 2} - 1 > 0$

Step 2

Simplify $\frac{2 x - 7}{x - 2} - 1$ .

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{2 x - 7}{x - 2} - 1 \cdot \frac{x - 2}{x - 2} > 0$

Combine $- 1$ and $\frac{x - 2}{x - 2}$ .

$\frac{2 x - 7}{x - 2} + \frac{- (x - 2)}{x - 2} > 0$

Combine the numerators over the common denominator.

$\frac{2 x - 7 - (x - 2)}{x - 2} > 0$

Simplify the numerator.

Apply the distributive property.

$\frac{2 x - 7 - x - - 2}{x - 2} > 0$

Multiply $- 1$ by $- 2$ .

$\frac{2 x - 7 - x + 2}{x - 2} > 0$

Subtract $x$ from $2 x$ .

$\frac{x - 7 + 2}{x - 2} > 0$

Add $- 7$ and $2$ .

$\frac{x - 5}{x - 2} > 0$

Step 3

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

$x - 5 = 0$

$x - 2 = 0$

Step 4

Add $5$ to both sides of the equation.

$x = 5$

Step 5

Add $2$ to both sides of the equation.

$x = 2$

Step 6

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

$x = 5$

$x = 2$

Step 7

Consolidate the solutions.

$x = 5, 2$

Step 8

Find the domain of $\frac{x - 5}{x - 2}$ .

Set the denominator in $\frac{x - 5}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

$(- \infty, 2) \cup (2, \infty)$

Step 9

Use each root to create test intervals.

$x < 2$

$2 < x < 5$

$x > 5$

Step 10

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 $x < 2$ 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.

$x = 0$

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

$\frac{2 (0) - 7}{(0) - 2} > 1$

The left side $3.5$ is greater than the right side $1$ , which means that the given statement is always true.

True

Test a value on the interval $2 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 5$ and see if this value makes the original inequality true.

$x = 4$

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

$\frac{2 (4) - 7}{(4) - 2} > 1$

The left side $0.5$ is not greater than the right side $1$ , which means that the given statement is false.

False

Test a value on the interval $x > 5$ to see if it makes the inequality true.

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

$\frac{2 (8) - 7}{(8) - 2} > 1$

The left side $1.5$ is greater than the right side $1$ , which means that the given statement is always true.

True

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

$x < 2$ True

$2 < x < 5$ False

$x > 5$ True

$x < 2$ True

$2 < x < 5$ False

$x > 5$ True

Step 11

The solution consists of all of the true intervals.

$x < 2$ or $x > 5$

Step 12

Convert the inequality to interval notation.

$(- \infty, 2) \cup (5, \infty)$

Step 13

Convert to Interval Notation (2x-7)/(x-2)>1

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