Solve the Inequality for x (5x)/(x-1)<4

$\frac{5 x}{x - 1} < 4$

Step 1

Subtract $4$ from both sides of the inequality.

$\frac{5 x}{x - 1} - 4 < 0$

Step 2

Simplify $\frac{5 x}{x - 1} - 4$ .

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

$\frac{5 x}{x - 1} - 4 \cdot \frac{x - 1}{x - 1} < 0$

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

$\frac{5 x}{x - 1} + \frac{- 4 (x - 1)}{x - 1} < 0$

Combine the numerators over the common denominator.

$\frac{5 x - 4 (x - 1)}{x - 1} < 0$

Simplify the numerator.

Apply the distributive property.

$\frac{5 x - 4 x - 4 \cdot - 1}{x - 1} < 0$

Multiply $- 4$ by $- 1$ .

$\frac{5 x - 4 x + 4}{x - 1} < 0$

Subtract $4 x$ from $5 x$ .

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

Step 3

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

$x + 4 = 0$

$x - 1 = 0$

Step 4

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 5

Add $1$ to both sides of the equation.

$x = 1$

Step 6

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

$x = - 4$

$x = 1$

Step 7

Consolidate the solutions.

$x = - 4, 1$

Step 8

Find the domain of $\frac{x + 4}{x - 1}$ .

Set the denominator in $\frac{x + 4}{x - 1}$ equal to $0$ to find where the expression is undefined.

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

$(- \infty, 1) \cup (1, \infty)$

Step 9

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 1$

$x > 1$

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 $x < - 4$ to see if it makes the inequality true.

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

$x = - 6$

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

$\frac{5 (- 6)}{(- 6) - 1} < 4$

The left side $4.\overline{285714}$ is not less than the right side $4$ , which means that the given statement is false.

False

Test a value on the interval $- 4 < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- 4 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

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

$\frac{5 (0)}{(0) - 1} < 4$

The left side $0$ is less than the right side $4$ , which means that the given statement is always true.

True

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.

$x = 4$

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

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

The left side $6.\overline{6}$ is not less than the right side $4$ , which means that the given statement is false.

False

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

$x < - 4$ False

$- 4 < x < 1$ True

$x > 1$ False

$x < - 4$ False

$- 4 < x < 1$ True

$x > 1$ False

Step 11

The solution consists of all of the true intervals.

$- 4 < x < 1$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- 4 < x < 1$

Interval Notation:

$(- 4, 1)$

Step 13

Solve the Inequality for x (5x)/(x-1)<4

Related Questions

Support LCMGCF.com by disabling your adblocker.