Solve the Inequality for x 4/(x+1)<4

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

Subtract $4$ from both sides of the inequality.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Factor $4$ out of $4 - 4 (x + 1)$ .

Factor $4$ out of $4$ .

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

Factor $4$ out of $- 4 (x + 1)$ .

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

Factor $4$ out of $4 (1) + 4 (- (x + 1))$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Subtract $1$ from $1$ .

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

Add $- x$ and $0$ .

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

Combine exponents.

Factor out negative.

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

Multiply $4$ by $- 1$ .

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

Move the negative in front of the fraction.

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

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

$- 4 x = 0$

$x + 1 = 0$

Divide each term in $- 4 x = 0$ by $- 4$ and simplify.

Divide each term in $- 4 x = 0$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{0}{- 4}$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{0}{- 4}$

Divide $x$ by $1$ .

$x = \frac{0}{- 4}$

Simplify the right side.

Divide $0$ by $- 4$ .

$x = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

$x = 0$

$x = - 1$

Consolidate the solutions.

$x = 0, - 1$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 0$

$x > 0$

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.

$x = - 4$

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

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

The left side $- 1.\overline{3}$ is less than the right side $4$ , which means that the given statement is always true.

True

Test a value on the interval $- 1 < x < 0$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 0$ and see if this value makes the original inequality true.

$x = - 0.5$

Replace $x$ with $- 0.5$ in the original inequality.

$\frac{4}{(- 0.5) + 1} < 4$

The left side $8$ is not less than the right side $4$ , which means that the given statement is false.

False

Test a value on the interval $x > 0$ to see if it makes the inequality true.

Choose a value on the interval $x > 0$ and see if this value makes the original inequality true.

$x = 2$

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

$\frac{4}{(2) + 1} < 4$

The left side $1.\overline{3}$ is less than the right side $4$ , which means that the given statement is always true.

True

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

$x < - 1$ True

$- 1 < x < 0$ False

$x > 0$ True

$x < - 1$ True

$- 1 < x < 0$ False

$x > 0$ True

The solution consists of all of the true intervals.

$x < - 1$ or $x > 0$

The result can be shown in multiple forms.

Inequality Form:

$x < - 1 or x > 0$

Interval Notation:

$(- \infty, - 1) \cup (0, \infty)$

Solve the Inequality for x 4/(x+1)<4

Related Questions

Support LCMGCF.com by disabling your adblocker.