Solve the Inequality for x 3/(x-4)<=-1

$\frac{3}{x - 4} \leq - 1$

Add $1$ to both sides of the inequality.

$\frac{3}{x - 4} + 1 \leq 0$

Simplify $\frac{3}{x - 4} + 1$ .

Write $1$ as a fraction with a common denominator.

$\frac{3}{x - 4} + \frac{x - 4}{x - 4} \leq 0$

Combine the numerators over the common denominator.

$\frac{3 + x - 4}{x - 4} \leq 0$

Subtract $4$ from $3$ .

$\frac{x - 1}{x - 4} \leq 0$

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

$x - 1 = 0$

$x - 4 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Add $4$ to both sides of the equation.

$x = 4$

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

$x = 1$

$x = 4$

Consolidate the solutions.

$x = 1, 4$

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

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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

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

Use each root to create test intervals.

$x < 1$

$1 < x < 4$

$x > 4$

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.

$x = 0$

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

$\frac{3}{(0) - 4} \leq - 1$

The left side $- 0.75$ is greater than the right side $- 1$ , which means that the given statement is false.

False

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

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

$x = 2$

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

$\frac{3}{(2) - 4} \leq - 1$

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

True

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.

$\frac{3}{(6) - 4} \leq - 1$

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

False

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

$x < 1$ False

$1 < x < 4$ True

$x > 4$ False

$x < 1$ False

$1 < x < 4$ True

$x > 4$ False

The solution consists of all of the true intervals.

$1 \leq x < 4$

The result can be shown in multiple forms.

Inequality Form:

$1 \leq x < 4$

Interval Notation:

$[1, 4)$

Solve the Inequality for x 3/(x-4)<=-1

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