Pre-Algebra Examples

$\frac{2}{x - 4} < 1$

Step 1

Subtract $1$ from both sides of the inequality.

$\frac{2}{x - 4} - 1 < 0$

Step 2

Simplify $\frac{2}{x - 4} - 1$ .

Tap for more steps...

Step 2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{2}{x - 4} - 1 \cdot \frac{x - 4}{x - 4} < 0$

Step 2.2

Combine $- 1$ and $\frac{x - 4}{x - 4}$ .

$\frac{2}{x - 4} + \frac{- (x - 4)}{x - 4} < 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{2 - (x - 4)}{x - 4} < 0$

Step 2.4

Simplify the numerator.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

$\frac{2 - x - - 4}{x - 4} < 0$

Step 2.4.2

Multiply $- 1$ by $- 4$ .

$\frac{2 - x + 4}{x - 4} < 0$

Step 2.4.3

Add $2$ and $4$ .

$\frac{- x + 6}{x - 4} < 0$

Step 2.5

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 6}{x - 4} < 0$

Step 2.6

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{- (x) - 1 (- 6)}{x - 4} < 0$

Step 2.7

Factor $- 1$ out of $- (x) - 1 (- 6)$ .

$\frac{- (x - 6)}{x - 4} < 0$

Step 2.8

Rewrite $- (x - 6)$ as $- 1 (x - 6)$ .

$\frac{- 1 (x - 6)}{x - 4} < 0$

Step 2.9

Move the negative in front of the fraction.

$- \frac{x - 6}{x - 4} < 0$

Step 3

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

$x - 6 = 0$

$x - 4 = 0$

Step 4

Add $6$ to both sides of the equation.

$x = 6$

Step 5

Add $4$ to both sides of the equation.

$x = 4$

Step 6

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

$x = 6$

$x = 4$

Step 7

Consolidate the solutions.

$x = 6, 4$

Step 8

Find the domain of $- \frac{x - 6}{x - 4}$ .

Tap for more steps...

Step 8.1

Set the denominator in $\frac{x - 6}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Step 8.2

Add $4$ to both sides of the equation.

$x = 4$

Step 8.3

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

$(- \infty, 4) \cup (4, \infty)$

Step 9

Use each root to create test intervals.

$x < 4$

$4 < x < 6$

$x > 6$

Step 10

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

Tap for more steps...

Step 10.1

Test a value on the interval $x < 4$ to see if it makes the inequality true.

Tap for more steps...

Step 10.1.1

Choose a value on the interval $x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 10.1.2

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

$\frac{2}{(0) - 4} < 1$

Step 10.1.3

The left side $- 0.5$ is less than the right side $1$ , which means that the given statement is always true.

True

Step 10.2

Test a value on the interval $4 < x < 6$ to see if it makes the inequality true.

Tap for more steps...

Step 10.2.1

Choose a value on the interval $4 < x < 6$ and see if this value makes the original inequality true.

$x = 5$

Step 10.2.2

Replace $x$ with $5$ in the original inequality.

$\frac{2}{(5) - 4} < 1$

Step 10.2.3

The left side $2$ is not less than the right side $1$ , which means that the given statement is false.

False

Step 10.3

Test a value on the interval $x > 6$ to see if it makes the inequality true.

Tap for more steps...

Step 10.3.1

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

Step 10.3.2

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

$\frac{2}{(8) - 4} < 1$

Step 10.3.3

The left side $0.5$ is less than the right side $1$ , which means that the given statement is always true.

True

Step 10.4

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

$x < 4$ True

$4 < x < 6$ False

$x > 6$ True

$x < 4$ True

$4 < x < 6$ False

$x > 6$ True

Step 11

The solution consists of all of the true intervals.

$x < 4$ or $x > 6$

Step 12