Convert to Interval Notation |5-7x|>9

$| 5 - 7 x | > 9$

Step 1

Write $| 5 - 7 x | > 9$ as a piecewise.

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

$5 - 7 x \geq 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$- 7 x \geq - 5$

Divide each term in $- 7 x \geq - 5$ by $- 7$ and simplify.

Divide each term in $- 7 x \geq - 5$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} \leq \frac{- 5}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} \leq \frac{- 5}{- 7}$

Divide $x$ by $1$ .

$x \leq \frac{- 5}{- 7}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \leq \frac{5}{7}$

In the piece where $5 - 7 x$ is non-negative, remove the absolute value.

$5 - 7 x > 9$

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

$5 - 7 x < 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$- 7 x < - 5$

Divide each term in $- 7 x < - 5$ by $- 7$ and simplify.

Divide each term in $- 7 x < - 5$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} > \frac{- 5}{- 7}$

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

$\frac{- 7 x}{- 7} > \frac{- 5}{- 7}$

Divide $x$ by $1$ .

$x > \frac{- 5}{- 7}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x > \frac{5}{7}$

In the piece where $5 - 7 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (5 - 7 x) > 9$

Write as a piecewise.

${\begin{cases} 5 - 7 x > 9 & x \leq \frac{5}{7} \\ - (5 - 7 x) > 9 & x > \frac{5}{7} \end{cases}$

Simplify $- (5 - 7 x) > 9$ .

Apply the distributive property.

${\begin{cases} 5 - 7 x > 9 & x \leq \frac{5}{7} \\ - 1 \cdot 5 - (- 7 x) > 9 & x > \frac{5}{7} \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} 5 - 7 x > 9 & x \leq \frac{5}{7} \\ - 5 - (- 7 x) > 9 & x > \frac{5}{7} \end{cases}$

Multiply $- 7$ by $- 1$ .

${\begin{cases} 5 - 7 x > 9 & x \leq \frac{5}{7} \\ - 5 + 7 x > 9 & x > \frac{5}{7} \end{cases}$

Step 2

Solve $5 - 7 x > 9$ for $x$ .

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

Subtract $5$ from both sides of the inequality.

$- 7 x > 9 - 5$

Subtract $5$ from $9$ .

$- 7 x > 4$

Divide each term in $- 7 x > 4$ by $- 7$ and simplify.

Divide each term in $- 7 x > 4$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Simplify the left side.

Cancel the common factor of $- 7$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 3

Solve $- 5 + 7 x > 9$ for $x$ .

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

Add $5$ to both sides of the inequality.

$7 x > 9 + 5$

Add $9$ and $5$ .

$7 x > 14$

Divide each term in $7 x > 14$ by $7$ and simplify.

Divide each term in $7 x > 14$ by $7$ .

$\frac{7 x}{7} > \frac{14}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} > \frac{14}{7}$

Divide $x$ by $1$ .

$x > \frac{14}{7}$

Simplify the right side.

Divide $14$ by $7$ .

$x > 2$

Step 4

Find the union of the solutions.

$x < - \frac{4}{7}$ or $x > 2$

Step 5

Convert the inequality to interval notation.

$(- \infty, - \frac{4}{7}) \cup (2, \infty)$

Step 6

Convert to Interval Notation |5-7x|>9

Related Questions

Support LCMGCF.com by disabling your adblocker.