Solve the Absolute Value Inequality for x |x-7|<8

$| x - 7 | < 8$

Write $| x - 7 | < 8$ as a piecewise.

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

$x - 7 \geq 0$

Add $7$ to both sides of the inequality.

$x \geq 7$

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

$x - 7 < 8$

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

$x - 7 < 0$

Add $7$ to both sides of the inequality.

$x < 7$

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

$- (x - 7) < 8$

Write as a piecewise.

${\begin{cases} x - 7 < 8 & x \geq 7 \\ - (x - 7) < 8 & x < 7 \end{cases}$

Simplify $- (x - 7) < 8$ .

Apply the distributive property.

${\begin{cases} x - 7 < 8 & x \geq 7 \\ - x - - 7 < 8 & x < 7 \end{cases}$

Multiply $- 1$ by $- 7$ .

${\begin{cases} x - 7 < 8 & x \geq 7 \\ - x + 7 < 8 & x < 7 \end{cases}$

Solve $x - 7 < 8$ when $x \geq 7$ .

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

Add $7$ to both sides of the inequality.

$x < 8 + 7$

Add $8$ and $7$ .

$x < 15$

Find the intersection of $x < 15$ and $x \geq 7$ .

$7 \leq x < 15$

Solve $- x + 7 < 8$ when $x < 7$ .

Solve $- x + 7 < 8$ for $x$ .

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

Subtract $7$ from both sides of the inequality.

$- x < 8 - 7$

Subtract $7$ from $8$ .

$- x < 1$

Divide each term in $- x < 1$ by $- 1$ and simplify.

Divide each term in $- x < 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{1}{- 1}$

Divide $x$ by $1$ .

$x > \frac{1}{- 1}$

Simplify the right side.

Divide $1$ by $- 1$ .

$x > - 1$

Find the intersection of $x > - 1$ and $x < 7$ .

$- 1 < x < 7$

Find the union of the solutions.

$- 1 < x < 15$

The result can be shown in multiple forms.

Inequality Form:

$- 1 < x < 15$

Interval Notation:

$(- 1, 15)$

Solve the Absolute Value Inequality for x |x-7|<8

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