Convert to Set Notation |(2-7x)/4|>=2/3

$| \frac{2 - 7 x}{4} | \geq \frac{2}{3}$

Step 1

Solve $x \leq - \frac{2}{21} or x \geq \frac{2}{3}$ .

Write $| \frac{2 - 7 x}{4} | \geq \frac{2}{3}$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$\frac{2 - 7 x}{4} \geq 0$

Solve the inequality.

Multiply both sides by $4$ .

$\frac{2 - 7 x}{4} \cdot 4 \geq 0 \cdot 4$

Simplify.

Simplify the left side.

Simplify $\frac{2 - 7 x}{4} \cdot 4$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{2 - 7 x}{4} \cdot 4 \geq 0 \cdot 4$

Rewrite the expression.

$2 - 7 x \geq 0 \cdot 4$

Reorder $2$ and $- 7 x$ .

$- 7 x + 2 \geq 0 \cdot 4$

Simplify the right side.

Multiply $0$ by $4$ .

$- 7 x + 2 \geq 0$

Solve for $x$ .

Subtract $2$ from both sides of the inequality.

$- 7 x \geq - 2$

Divide each term in $- 7 x \geq - 2$ by $- 7$ and simplify.

Divide each term in $- 7 x \geq - 2$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \leq \frac{- 2}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} \leq \frac{- 2}{- 7}$

Divide $x$ by $1$ .

$x \leq \frac{- 2}{- 7}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \leq \frac{2}{7}$

In the piece where $\frac{2 - 7 x}{4}$ is non-negative, remove the absolute value.

$\frac{2 - 7 x}{4} \geq \frac{2}{3}$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$\frac{2 - 7 x}{4} < 0$

Solve the inequality.

Multiply both sides by $4$ .

$\frac{2 - 7 x}{4} \cdot 4 < 0 \cdot 4$

Simplify.

Simplify the left side.

Simplify $\frac{2 - 7 x}{4} \cdot 4$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{2 - 7 x}{4} \cdot 4 < 0 \cdot 4$

Rewrite the expression.

$2 - 7 x < 0 \cdot 4$

Reorder $2$ and $- 7 x$ .

$- 7 x + 2 < 0 \cdot 4$

Simplify the right side.

Multiply $0$ by $4$ .

$- 7 x + 2 < 0$

Solve for $x$ .

Subtract $2$ from both sides of the inequality.

$- 7 x < - 2$

Divide each term in $- 7 x < - 2$ by $- 7$ and simplify.

Divide each term in $- 7 x < - 2$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Dividing two negative values results in a positive value.

$x > \frac{2}{7}$

In the piece where $\frac{2 - 7 x}{4}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{2 - 7 x}{4} \geq \frac{2}{3}$

Write as a piecewise.

${\begin{cases} \frac{2 - 7 x}{4} \geq \frac{2}{3} & x \leq \frac{2}{7} \\ - \frac{2 - 7 x}{4} \geq \frac{2}{3} & x > \frac{2}{7} \end{cases}$

Solve $\frac{2 - 7 x}{4} \geq \frac{2}{3}$ for $x$ .

Multiply both sides by $4$ .

$\frac{2 - 7 x}{4} \cdot 4 \geq \frac{2}{3} \cdot 4$

Simplify.

Simplify the left side.

Simplify $\frac{2 - 7 x}{4} \cdot 4$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{2 - 7 x}{4} \cdot 4 \geq \frac{2}{3} \cdot 4$

Rewrite the expression.

$2 - 7 x \geq \frac{2}{3} \cdot 4$

Reorder $2$ and $- 7 x$ .

$- 7 x + 2 \geq \frac{2}{3} \cdot 4$

Simplify the right side.

Multiply $\frac{2}{3} \cdot 4$ .

Combine $\frac{2}{3}$ and $4$ .

$- 7 x + 2 \geq \frac{2 \cdot 4}{3}$

Multiply $2$ by $4$ .

$- 7 x + 2 \geq \frac{8}{3}$

Solve for $x$ .

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

Subtract $2$ from both sides of the inequality.

$- 7 x \geq \frac{8}{3} - 2$

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

$- 7 x \geq \frac{8}{3} - 2 \cdot \frac{3}{3}$

Combine $- 2$ and $\frac{3}{3}$ .

$- 7 x \geq \frac{8}{3} + \frac{- 2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$- 7 x \geq \frac{8 - 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 2$ by $3$ .

$- 7 x \geq \frac{8 - 6}{3}$

Subtract $6$ from $8$ .

$- 7 x \geq \frac{2}{3}$

Divide each term in $- 7 x \geq \frac{2}{3}$ by $- 7$ and simplify.

Divide each term in $- 7 x \geq \frac{2}{3}$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \leq \frac{\frac{2}{3}}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} \leq \frac{\frac{2}{3}}{- 7}$

Divide $x$ by $1$ .

$x \leq \frac{\frac{2}{3}}{- 7}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Move the negative in front of the fraction.

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

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

Multiply $\frac{2}{3}$ by $\frac{1}{7}$ .

$x \leq - \frac{2}{3 \cdot 7}$

Multiply $3$ by $7$ .

$x \leq - \frac{2}{21}$

Solve $- \frac{2 - 7 x}{4} \geq \frac{2}{3}$ for $x$ .

Divide each term in $- \frac{2 - 7 x}{4} \geq \frac{2}{3}$ by $- 1$ and simplify.

Divide each term in $- \frac{2 - 7 x}{4} \geq \frac{2}{3}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{2 - 7 x}{4}}{- 1} \leq \frac{\frac{2}{3}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{2 - 7 x}{4}}{1} \leq \frac{\frac{2}{3}}{- 1}$

Divide $\frac{2 - 7 x}{4}$ by $1$ .

$\frac{2 - 7 x}{4} \leq \frac{\frac{2}{3}}{- 1}$

Simplify the right side.

Move the negative one from the denominator of $\frac{\frac{2}{3}}{- 1}$ .

$\frac{2 - 7 x}{4} \leq - 1 \cdot \frac{2}{3}$

Rewrite $- 1 \cdot \frac{2}{3}$ as $- \frac{2}{3}$ .

$\frac{2 - 7 x}{4} \leq - \frac{2}{3}$

Multiply both sides by $4$ .

$\frac{2 - 7 x}{4} \cdot 4 \leq - \frac{2}{3} \cdot 4$

Simplify.

Simplify the left side.

Simplify $\frac{2 - 7 x}{4} \cdot 4$ .

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{2 - 7 x}{4} \cdot 4 \leq - \frac{2}{3} \cdot 4$

Rewrite the expression.

$2 - 7 x \leq - \frac{2}{3} \cdot 4$

Reorder $2$ and $- 7 x$ .

$- 7 x + 2 \leq - \frac{2}{3} \cdot 4$

Simplify the right side.

Simplify $- \frac{2}{3} \cdot 4$ .

Multiply $- \frac{2}{3} \cdot 4$ .

Multiply $4$ by $- 1$ .

$- 7 x + 2 \leq - 4 (\frac{2}{3})$

Combine $- 4$ and $\frac{2}{3}$ .

$- 7 x + 2 \leq \frac{- 4 \cdot 2}{3}$

Multiply $- 4$ by $2$ .

$- 7 x + 2 \leq \frac{- 8}{3}$

Move the negative in front of the fraction.

$- 7 x + 2 \leq - \frac{8}{3}$

Solve for $x$ .

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

Subtract $2$ from both sides of the inequality.

$- 7 x \leq - \frac{8}{3} - 2$

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

$- 7 x \leq - \frac{8}{3} - 2 \cdot \frac{3}{3}$

Combine $- 2$ and $\frac{3}{3}$ .

$- 7 x \leq - \frac{8}{3} + \frac{- 2 \cdot 3}{3}$

Combine the numerators over the common denominator.

$- 7 x \leq \frac{- 8 - 2 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 2$ by $3$ .

$- 7 x \leq \frac{- 8 - 6}{3}$

Subtract $6$ from $- 8$ .

$- 7 x \leq \frac{- 14}{3}$

Move the negative in front of the fraction.

$- 7 x \leq - \frac{14}{3}$

Divide each term in $- 7 x \leq - \frac{14}{3}$ by $- 7$ and simplify.

Divide each term in $- 7 x \leq - \frac{14}{3}$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \geq \frac{- \frac{14}{3}}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} \geq \frac{- \frac{14}{3}}{- 7}$

Divide $x$ by $1$ .

$x \geq \frac{- \frac{14}{3}}{- 7}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x \geq - \frac{14}{3} \cdot \frac{1}{- 7}$

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{14}{3}$ into the numerator.

$x \geq \frac{- 14}{3} \cdot \frac{1}{- 7}$

Factor $7$ out of $- 14$ .

$x \geq \frac{7 (- 2)}{3} \cdot \frac{1}{- 7}$

Factor $7$ out of $- 7$ .

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

Cancel the common factor.

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

Rewrite the expression.

$x \geq \frac{- 2}{3} \cdot \frac{1}{- 1}$

Multiply $\frac{- 2}{3}$ by $\frac{1}{- 1}$ .

$x \geq \frac{- 2}{3 \cdot - 1}$

Multiply $3$ by $- 1$ .

$x \geq \frac{- 2}{- 3}$

Dividing two negative values results in a positive value.

$x \geq \frac{2}{3}$

Find the union of the solutions.

$x \leq - \frac{2}{21}$ or $x \geq \frac{2}{3}$

Step 2

Use the inequality $x \leq - \frac{2}{21} or x \geq \frac{2}{3}$ to build the set notation.

${x | x \leq - \frac{2}{21} or x \geq \frac{2}{3}}$

Step 3

Convert to Set Notation |(2-7x)/4|>=2/3

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