Solve the Inequality for x |x-4|<9

$| x - 4 | < 9$

Step 1

Write $| x - 4 | < 9$ as a piecewise.

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

$x - 4 \geq 0$

Add $4$ to both sides of the inequality.

$x \geq 4$

In the piece where $x - 4$ is non-negative, remove the absolute value.

$x - 4 < 9$

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

$x - 4 < 0$

Add $4$ to both sides of the inequality.

$x < 4$

In the piece where $x - 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 4) < 9$

Write as a piecewise.

${\begin{cases} x - 4 < 9 & x \geq 4 \\ - (x - 4) < 9 & x < 4 \end{cases}$

Simplify $- (x - 4) < 9$ .

Apply the distributive property.

${\begin{cases} x - 4 < 9 & x \geq 4 \\ - x - - 4 < 9 & x < 4 \end{cases}$

Multiply $- 1$ by $- 4$ .

${\begin{cases} x - 4 < 9 & x \geq 4 \\ - x + 4 < 9 & x < 4 \end{cases}$

Step 2

Solve $x - 4 < 9$ when $x \geq 4$ .

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

Add $4$ to both sides of the inequality.

$x < 9 + 4$

Add $9$ and $4$ .

$x < 13$

Find the intersection of $x < 13$ and $x \geq 4$ .

$4 \leq x < 13$

Step 3

Solve $- x + 4 < 9$ when $x < 4$ .

Solve $- x + 4 < 9$ for $x$ .

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

Subtract $4$ from both sides of the inequality.

$- x < 9 - 4$

Subtract $4$ from $9$ .

$- x < 5$

Divide each term in $- x < 5$ by $- 1$ and simplify.

Divide each term in $- x < 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{5}{- 1}$

Divide $x$ by $1$ .

$x > \frac{5}{- 1}$

Simplify the right side.

Divide $5$ by $- 1$ .

$x > - 5$

Find the intersection of $x > - 5$ and $x < 4$ .

$- 5 < x < 4$

Step 4

Find the union of the solutions.

$- 5 < x < 13$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 5 < x < 13$

Interval Notation:

$(- 5, 13)$

Step 6

Solve the Inequality for x |x-4|<9

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