Solve for x |7x-7|+2<23

$| 7 x - 7 | + 2 < 23$

Step 1

Write $| 7 x - 7 | + 2 < 23$ as a piecewise.

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

$7 x - 7 \geq 0$

Solve the inequality.

Add $7$ to both sides of the inequality.

$7 x \geq 7$

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

Divide each term in $7 x \geq 7$ by $7$ .

$\frac{7 x}{7} \geq \frac{7}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} \geq \frac{7}{7}$

Divide $x$ by $1$ .

$x \geq \frac{7}{7}$

Simplify the right side.

Divide $7$ by $7$ .

$x \geq 1$

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

$7 x - 7 + 2 < 23$

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

$7 x - 7 < 0$

Solve the inequality.

Add $7$ to both sides of the inequality.

$7 x < 7$

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

Divide each term in $7 x < 7$ by $7$ .

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

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $7$ by $7$ .

$x < 1$

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

$- (7 x - 7) + 2 < 23$

Write as a piecewise.

${\begin{cases} 7 x - 7 + 2 < 23 & x \geq 1 \\ - (7 x - 7) + 2 < 23 & x < 1 \end{cases}$

Add $- 7$ and $2$ .

${\begin{cases} 7 x - 5 < 23 & x \geq 1 \\ - (7 x - 7) + 2 < 23 & x < 1 \end{cases}$

Simplify $- (7 x - 7) + 2 < 23$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 7 x - 5 < 23 & x \geq 1 \\ - (7 x) - - 7 + 2 < 23 & x < 1 \end{cases}$

Multiply $7$ by $- 1$ .

${\begin{cases} 7 x - 5 < 23 & x \geq 1 \\ - 7 x - - 7 + 2 < 23 & x < 1 \end{cases}$

Multiply $- 1$ by $- 7$ .

${\begin{cases} 7 x - 5 < 23 & x \geq 1 \\ - 7 x + 7 + 2 < 23 & x < 1 \end{cases}$

Add $7$ and $2$ .

${\begin{cases} 7 x - 5 < 23 & x \geq 1 \\ - 7 x + 9 < 23 & x < 1 \end{cases}$

Step 2

Solve $7 x - 5 < 23$ when $x \geq 1$ .

Solve $7 x - 5 < 23$ for $x$ .

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

Add $5$ to both sides of the inequality.

$7 x < 23 + 5$

Add $23$ and $5$ .

$7 x < 28$

Divide each term in $7 x < 28$ by $7$ and simplify.

Divide each term in $7 x < 28$ by $7$ .

$\frac{7 x}{7} < \frac{28}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x}{7} < \frac{28}{7}$

Divide $x$ by $1$ .

$x < \frac{28}{7}$

Simplify the right side.

Divide $28$ by $7$ .

$x < 4$

Find the intersection of $x < 4$ and $x \geq 1$ .

$1 \leq x < 4$

Step 3

Solve $- 7 x + 9 < 23$ when $x < 1$ .

Solve $- 7 x + 9 < 23$ for $x$ .

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

Subtract $9$ from both sides of the inequality.

$- 7 x < 23 - 9$

Subtract $9$ from $23$ .

$- 7 x < 14$

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

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

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

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 $- 7$ .

$x > - 2$

Find the intersection of $x > - 2$ and $x < 1$ .

$- 2 < x < 1$

Step 4

Find the union of the solutions.

$- 2 < x < 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 4$

Interval Notation:

$(- 2, 4)$

Step 6

Solve for x |7x-7|+2<23

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