Convert to Set Notation 1/(3x)-2/(7x)>1/3

$\frac{1}{3 x} - \frac{2}{7 x} > \frac{1}{3}$

Step 1

Solve $0 < x < \frac{1}{7}$ .

Simplify $\frac{1}{3 x} - \frac{2}{7 x}$ .

To write $\frac{1}{3 x}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{1}{3 x} \cdot \frac{7}{7} - \frac{2}{7 x} > \frac{1}{3}$

To write $- \frac{2}{7 x}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3 x} \cdot \frac{7}{7} - \frac{2}{7 x} \cdot \frac{3}{3} > \frac{1}{3}$

Write each expression with a common denominator of $21 x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{3 x}$ by $\frac{7}{7}$ .

$\frac{7}{3 x \cdot 7} - \frac{2}{7 x} \cdot \frac{3}{3} > \frac{1}{3}$

Multiply $7$ by $3$ .

$\frac{7}{21 x} - \frac{2}{7 x} \cdot \frac{3}{3} > \frac{1}{3}$

Multiply $\frac{2}{7 x}$ by $\frac{3}{3}$ .

$\frac{7}{21 x} - \frac{2 \cdot 3}{7 x \cdot 3} > \frac{1}{3}$

Multiply $3$ by $7$ .

$\frac{7}{21 x} - \frac{2 \cdot 3}{21 x} > \frac{1}{3}$

Combine the numerators over the common denominator.

$\frac{7 - 2 \cdot 3}{21 x} > \frac{1}{3}$

Simplify the numerator.

Multiply $- 2$ by $3$ .

$\frac{7 - 6}{21 x} > \frac{1}{3}$

Subtract $6$ from $7$ .

$\frac{1}{21 x} > \frac{1}{3}$

Multiply both sides by $21 x$ .

$\frac{1}{21 x} (21 x) = \frac{1}{3} (21 x)$

Simplify.

Simplify the left side.

Simplify $\frac{1}{21 x} (21 x)$ .

Rewrite using the commutative property of multiplication.

$21 \frac{1}{21 x} x = \frac{1}{3} (21 x)$

Cancel the common factor of $21$ .

Factor $21$ out of $21 x$ .

$21 \frac{1}{21 (x)} x = \frac{1}{3} (21 x)$

Cancel the common factor.

$21 \frac{1}{21 x} x = \frac{1}{3} (21 x)$

Rewrite the expression.

$\frac{1}{x} x = \frac{1}{3} (21 x)$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{1}{x} x = \frac{1}{3} (21 x)$

Rewrite the expression.

$1 = \frac{1}{3} (21 x)$

Simplify the right side.

Cancel the common factor of $3$ .

Factor $3$ out of $21 x$ .

$1 = \frac{1}{3} (3 (7 x))$

Cancel the common factor.

$1 = \frac{1}{3} (3 (7 x))$

Rewrite the expression.

$1 = 7 x$

Solve for $x$ .

Rewrite the equation as $7 x = 1$ .

$7 x = 1$

Divide each term in $7 x = 1$ by $7$ and simplify.

Divide each term in $7 x = 1$ by $7$ .

$\frac{7 x}{7} = \frac{1}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} = \frac{1}{7}$

Divide $x$ by $1$ .

$x = \frac{1}{7}$

Find the domain of $\frac{1}{21 x} - \frac{1}{3}$ .

Set the denominator in $\frac{1}{21 x}$ equal to $0$ to find where the expression is undefined.

$21 x = 0$

Divide each term in $21 x = 0$ by $21$ and simplify.

Divide each term in $21 x = 0$ by $21$ .

$\frac{21 x}{21} = \frac{0}{21}$

Simplify the left side.

Cancel the common factor of $21$ .

Cancel the common factor.

$\frac{21 x}{21} = \frac{0}{21}$

Divide $x$ by $1$ .

$x = \frac{0}{21}$

Simplify the right side.

Divide $0$ by $21$ .

$x = 0$

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

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

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{1}{7}$

$x > \frac{1}{7}$

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.

$\frac{1}{3 (- 2)} - \frac{2}{7 (- 2)} > \frac{1}{3}$

The left side $- 0.0\overline{238095}$ is not greater than the right side $0.\overline{3}$ , which means that the given statement is false.

False

Test a value on the interval $0 < x < \frac{1}{7}$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < \frac{1}{7}$ and see if this value makes the original inequality true.

$x = 0.07$

Replace $x$ with $0.07$ in the original inequality.

$\frac{1}{3 (0.07)} - \frac{2}{7 (0.07)} > \frac{1}{3}$

The left side $0.6802721$ is greater than the right side $0.\overline{3}$ , which means that the given statement is always true.

True

Test a value on the interval $x > \frac{1}{7}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{1}{7}$ and see if this value makes the original inequality true.

$x = 3$

Replace $x$ with $3$ in the original inequality.

$\frac{1}{3 (3)} - \frac{2}{7 (3)} > \frac{1}{3}$

The left side $0.\overline{015873}$ is not greater than the right side $0.\overline{3}$ , which means that the given statement is false.

False

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

$x < 0$ False

$0 < x < \frac{1}{7}$ True

$x > \frac{1}{7}$ False

$x < 0$ False

$0 < x < \frac{1}{7}$ True

$x > \frac{1}{7}$ False

The solution consists of all of the true intervals.

$0 < x < \frac{1}{7}$

Step 2

Use the inequality $0 < x < \frac{1}{7}$ to build the set notation.

${x | 0 < x < \frac{1}{7}}$

Step 3

Convert to Set Notation 1/(3x)-2/(7x)>1/3

Related Questions

Support LCMGCF.com by disabling your adblocker.