Solve the Absolute Value Inequality for x |x-2|-5<-2

$| x - 2 | - 5 < - 2$

Step 1

Write $| x - 2 | - 5 < - 2$ as a piecewise.

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

$x - 2 \geq 0$

Add $2$ to both sides of the inequality.

$x \geq 2$

In the piece where $x - 2$ is non-negative, remove the absolute value.

$x - 2 - 5 < - 2$

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

$x - 2 < 0$

Add $2$ to both sides of the inequality.

$x < 2$

In the piece where $x - 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 2) - 5 < - 2$

Write as a piecewise.

${\begin{cases} x - 2 - 5 < - 2 & x \geq 2 \\ - (x - 2) - 5 < - 2 & x < 2 \end{cases}$

Subtract $5$ from $- 2$ .

${\begin{cases} x - 7 < - 2 & x \geq 2 \\ - (x - 2) - 5 < - 2 & x < 2 \end{cases}$

Simplify $- (x - 2) - 5 < - 2$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} x - 7 < - 2 & x \geq 2 \\ - x - - 2 - 5 < - 2 & x < 2 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} x - 7 < - 2 & x \geq 2 \\ - x + 2 - 5 < - 2 & x < 2 \end{cases}$

Subtract $5$ from $2$ .

${\begin{cases} x - 7 < - 2 & x \geq 2 \\ - x - 3 < - 2 & x < 2 \end{cases}$

Step 2

Solve $x - 7 < - 2$ when $x \geq 2$ .

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

Add $7$ to both sides of the inequality.

$x < - 2 + 7$

Add $- 2$ and $7$ .

$x < 5$

Find the intersection of $x < 5$ and $x \geq 2$ .

$2 \leq x < 5$

Step 3

Solve $- x - 3 < - 2$ when $x < 2$ .

Solve $- x - 3 < - 2$ for $x$ .

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

Add $3$ to both sides of the inequality.

$- x < - 2 + 3$

Add $- 2$ and $3$ .

$- x < 1$

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 $x < 2$ .

$- 1 < x < 2$

Step 4

Find the union of the solutions.

$- 1 < x < 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 1 < x < 5$

Interval Notation:

$(- 1, 5)$

Step 6

Solve the Absolute Value Inequality for x |x-2|-5<-2

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