Solve the Absolute Value Inequality for x 9|x+8|+10<55

$9 | x + 8 | + 10 < 55$

Step 1

Write $9 | x + 8 | + 10 < 55$ as a piecewise.

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

$x + 8 \geq 0$

Subtract $8$ from both sides of the inequality.

$x \geq - 8$

In the piece where $x + 8$ is non-negative, remove the absolute value.

$9 (x + 8) + 10 < 55$

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

$x + 8 < 0$

Subtract $8$ from both sides of the inequality.

$x < - 8$

In the piece where $x + 8$ is negative, remove the absolute value and multiply by $- 1$ .

$9 (- (x + 8)) + 10 < 55$

Write as a piecewise.

${\begin{cases} 9 (x + 8) + 10 < 55 & x \geq - 8 \\ 9 (- (x + 8)) + 10 < 55 & x < - 8 \end{cases}$

Simplify $9 (x + 8) + 10 < 55$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 9 x + 9 \cdot 8 + 10 < 55 & x \geq - 8 \\ 9 (- (x + 8)) + 10 < 55 & x < - 8 \end{cases}$

Multiply $9$ by $8$ .

${\begin{cases} 9 x + 72 + 10 < 55 & x \geq - 8 \\ 9 (- (x + 8)) + 10 < 55 & x < - 8 \end{cases}$

Add $72$ and $10$ .

${\begin{cases} 9 x + 82 < 55 & x \geq - 8 \\ 9 (- (x + 8)) + 10 < 55 & x < - 8 \end{cases}$

Simplify $9 (- (x + 8)) + 10 < 55$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 9 x + 82 < 55 & x \geq - 8 \\ 9 (- x - 1 \cdot 8) + 10 < 55 & x < - 8 \end{cases}$

Multiply $- 1$ by $8$ .

${\begin{cases} 9 x + 82 < 55 & x \geq - 8 \\ 9 (- x - 8) + 10 < 55 & x < - 8 \end{cases}$

Apply the distributive property.

${\begin{cases} 9 x + 82 < 55 & x \geq - 8 \\ 9 (- x) + 9 \cdot - 8 + 10 < 55 & x < - 8 \end{cases}$

Multiply $- 1$ by $9$ .

${\begin{cases} 9 x + 82 < 55 & x \geq - 8 \\ - 9 x + 9 \cdot - 8 + 10 < 55 & x < - 8 \end{cases}$

Multiply $9$ by $- 8$ .

${\begin{cases} 9 x + 82 < 55 & x \geq - 8 \\ - 9 x - 72 + 10 < 55 & x < - 8 \end{cases}$

Add $- 72$ and $10$ .

${\begin{cases} 9 x + 82 < 55 & x \geq - 8 \\ - 9 x - 62 < 55 & x < - 8 \end{cases}$

Step 2

Solve $9 x + 82 < 55$ when $x \geq - 8$ .

Solve $9 x + 82 < 55$ for $x$ .

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

Subtract $82$ from both sides of the inequality.

$9 x < 55 - 82$

Subtract $82$ from $55$ .

$9 x < - 27$

Divide each term in $9 x < - 27$ by $9$ and simplify.

Divide each term in $9 x < - 27$ by $9$ .

$\frac{9 x}{9} < \frac{- 27}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} < \frac{- 27}{9}$

Divide $x$ by $1$ .

$x < \frac{- 27}{9}$

Simplify the right side.

Divide $- 27$ by $9$ .

$x < - 3$

Find the intersection of $x < - 3$ and $x \geq - 8$ .

$- 8 \leq x < - 3$

Step 3

Solve $- 9 x - 62 < 55$ when $x < - 8$ .

Solve $- 9 x - 62 < 55$ for $x$ .

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

Add $62$ to both sides of the inequality.

$- 9 x < 55 + 62$

Add $55$ and $62$ .

$- 9 x < 117$

Divide each term in $- 9 x < 117$ by $- 9$ and simplify.

Divide each term in $- 9 x < 117$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 x}{- 9} > \frac{117}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 x}{- 9} > \frac{117}{- 9}$

Divide $x$ by $1$ .

$x > \frac{117}{- 9}$

Simplify the right side.

Divide $117$ by $- 9$ .

$x > - 13$

Find the intersection of $x > - 13$ and $x < - 8$ .

$- 13 < x < - 8$

Step 4

Find the union of the solutions.

$- 13 < x < - 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 13 < x < - 3$

Interval Notation:

$(- 13, - 3)$

Step 6

Solve the Absolute Value Inequality for x 9|x+8|+10<55

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