Solve for x |x-1|<5

$| x - 1 | < 5$

Step 1

Write $| x - 1 | < 5$ as a piecewise.

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

$x - 1 \geq 0$

Add $1$ to both sides of the inequality.

$x \geq 1$

In the piece where $x - 1$ is non-negative, remove the absolute value.

$x - 1 < 5$

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

$x - 1 < 0$

Add $1$ to both sides of the inequality.

$x < 1$

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

$- (x - 1) < 5$

Write as a piecewise.

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

Simplify $- (x - 1) < 5$ .

Apply the distributive property.

${\begin{cases} x - 1 < 5 & x \geq 1 \\ - x - - 1 < 5 & x < 1 \end{cases}$

Multiply $- 1$ by $- 1$ .

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

Step 2

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

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

Add $1$ to both sides of the inequality.

$x < 5 + 1$

Add $5$ and $1$ .

$x < 6$

Find the intersection of $x < 6$ and $x \geq 1$ .

$1 \leq x < 6$

Step 3

Solve $- x + 1 < 5$ when $x < 1$ .

Solve $- x + 1 < 5$ for $x$ .

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

Subtract $1$ from both sides of the inequality.

$- x < 5 - 1$

Subtract $1$ from $5$ .

$- x < 4$

Divide each term in $- x < 4$ by $- 1$ and simplify.

Divide each term in $- x < 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{4}{- 1}$

Divide $x$ by $1$ .

$x > \frac{4}{- 1}$

Simplify the right side.

Divide $4$ by $- 1$ .

$x > - 4$

Find the intersection of $x > - 4$ and $x < 1$ .

$- 4 < x < 1$

Step 4

Find the union of the solutions.

$- 4 < x < 6$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 4 < x < 6$

Interval Notation:

$(- 4, 6)$

Step 6

Solve for x |x-1|<5

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