Solve for x |x-5|<4

$| x - 5 | < 4$

Step 1

Write $| x - 5 | < 4$ as a piecewise.

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

$x - 5 \geq 0$

Add $5$ to both sides of the inequality.

$x \geq 5$

In the piece where $x - 5$ is non-negative, remove the absolute value.

$x - 5 < 4$

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

$x - 5 < 0$

Add $5$ to both sides of the inequality.

$x < 5$

In the piece where $x - 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 5) < 4$

Write as a piecewise.

${\begin{cases} x - 5 < 4 & x \geq 5 \\ - (x - 5) < 4 & x < 5 \end{cases}$

Simplify $- (x - 5) < 4$ .

Apply the distributive property.

${\begin{cases} x - 5 < 4 & x \geq 5 \\ - x - - 5 < 4 & x < 5 \end{cases}$

Multiply $- 1$ by $- 5$ .

${\begin{cases} x - 5 < 4 & x \geq 5 \\ - x + 5 < 4 & x < 5 \end{cases}$

Step 2

Solve $x - 5 < 4$ when $x \geq 5$ .

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

Add $5$ to both sides of the inequality.

$x < 4 + 5$

Add $4$ and $5$ .

$x < 9$

Find the intersection of $x < 9$ and $x \geq 5$ .

$5 \leq x < 9$

Step 3

Solve $- x + 5 < 4$ when $x < 5$ .

Solve $- x + 5 < 4$ for $x$ .

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

Subtract $5$ from both sides of the inequality.

$- x < 4 - 5$

Subtract $5$ from $4$ .

$- x < - 1$

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 $x < 5$ .

$1 < x < 5$

Step 4

Find the union of the solutions.

$1 < x < 9$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$1 < x < 9$

Interval Notation:

$(1, 9)$

Step 6

Solve for x |x-5|<4

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