Solve for x |7-9x|>2

$| 7 - 9 x | > 2$

Step 1

Write $| 7 - 9 x | > 2$ as a piecewise.

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

$7 - 9 x \geq 0$

Solve the inequality.

Subtract $7$ from both sides of the inequality.

$- 9 x \geq - 7$

Divide each term in $- 9 x \geq - 7$ by $- 9$ and simplify.

Divide each term in $- 9 x \geq - 7$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 x}{- 9} \leq \frac{- 7}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 x}{- 9} \leq \frac{- 7}{- 9}$

Divide $x$ by $1$ .

$x \leq \frac{- 7}{- 9}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x \leq \frac{7}{9}$

In the piece where $7 - 9 x$ is non-negative, remove the absolute value.

$7 - 9 x > 2$

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

$7 - 9 x < 0$

Solve the inequality.

Subtract $7$ from both sides of the inequality.

$- 9 x < - 7$

Divide each term in $- 9 x < - 7$ by $- 9$ and simplify.

Divide each term in $- 9 x < - 7$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 x}{- 9} > \frac{- 7}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 x}{- 9} > \frac{- 7}{- 9}$

Divide $x$ by $1$ .

$x > \frac{- 7}{- 9}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x > \frac{7}{9}$

In the piece where $7 - 9 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- (7 - 9 x) > 2$

Write as a piecewise.

${\begin{cases} 7 - 9 x > 2 & x \leq \frac{7}{9} \\ - (7 - 9 x) > 2 & x > \frac{7}{9} \end{cases}$

Simplify $- (7 - 9 x) > 2$ .

Apply the distributive property.

${\begin{cases} 7 - 9 x > 2 & x \leq \frac{7}{9} \\ - 1 \cdot 7 - (- 9 x) > 2 & x > \frac{7}{9} \end{cases}$

Multiply $- 1$ by $7$ .

${\begin{cases} 7 - 9 x > 2 & x \leq \frac{7}{9} \\ - 7 - (- 9 x) > 2 & x > \frac{7}{9} \end{cases}$

Multiply $- 9$ by $- 1$ .

${\begin{cases} 7 - 9 x > 2 & x \leq \frac{7}{9} \\ - 7 + 9 x > 2 & x > \frac{7}{9} \end{cases}$

Step 2

Solve $7 - 9 x > 2$ for $x$ .

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

Subtract $7$ from both sides of the inequality.

$- 9 x > 2 - 7$

Subtract $7$ from $2$ .

$- 9 x > - 5$

Divide each term in $- 9 x > - 5$ by $- 9$ and simplify.

Divide each term in $- 9 x > - 5$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 x}{- 9} < \frac{- 5}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 x}{- 9} < \frac{- 5}{- 9}$

Divide $x$ by $1$ .

$x < \frac{- 5}{- 9}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x < \frac{5}{9}$

Step 3

Solve $- 7 + 9 x > 2$ for $x$ .

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

Add $7$ to both sides of the inequality.

$9 x > 2 + 7$

Add $2$ and $7$ .

$9 x > 9$

Divide each term in $9 x > 9$ by $9$ and simplify.

Divide each term in $9 x > 9$ by $9$ .

$\frac{9 x}{9} > \frac{9}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} > \frac{9}{9}$

Divide $x$ by $1$ .

$x > \frac{9}{9}$

Simplify the right side.

Divide $9$ by $9$ .

$x > 1$

Step 4

Find the union of the solutions.

$x < \frac{5}{9}$ or $x > 1$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{5}{9} or x > 1$

Interval Notation:

$(- \infty, \frac{5}{9}) \cup (1, \infty)$

Step 6

Solve for x |7-9x|>2

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