Solve the Inequality for x 1/x+3>4

$\frac{1}{x} + 3 > 4$

Move all terms not containing $x$ to the right side of the inequality.

Subtract $3$ from both sides of the inequality.

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

Subtract $3$ from $4$ .

$\frac{1}{x} > 1$

Multiply both sides by $x$ .

$\frac{1}{x} x = 1 x$

Simplify.

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{1}{x} x = 1 x$

Rewrite the expression.

$1 = 1 x$

Simplify the right side.

Multiply $x$ by $1$ .

$1 = x$

Rewrite the equation as $x = 1$ .

$x = 1$

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

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

$x = 0$

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

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

Use each root to create test intervals.

$x < 0$

$0 < x < 1$

$x > 1$

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 $x < 0$ 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.

$\frac{1}{- 2} + 3 > 4$

The left side $2.5$ is not greater than the right side $4$ , which means that the given statement is false.

False

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

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

$x = 0.5$

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

$\frac{1}{0.5} + 3 > 4$

The left side $5$ is greater 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.

$\frac{1}{4} + 3 > 4$

The left side $3.25$ is not greater 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.

$x < 0$ False

$0 < x < 1$ True

$x > 1$ False

$x < 0$ False

$0 < x < 1$ True

$x > 1$ False

The solution consists of all of the true intervals.

$0 < x < 1$

The result can be shown in multiple forms.

Inequality Form:

$0 < x < 1$

Interval Notation:

$(0, 1)$

Solve the Inequality for x 1/x+3>4

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