Solve for x |-2+(x-3)|

$| - 2 + (x - 3) | < 7$

Write $| - 2 + x - 3 | < 7$ as a piecewise.

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

$- 2 + x - 3 \geq 0$

Solve the inequality.

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Subtract $3$ from $- 2$ .

$x - 5 \geq 0$

Add $5$ to both sides of the inequality.

$x \geq 5$

In the piece where $- 2 + x - 3$ is non-negative, remove the absolute value.

$- 2 + x - 3 < 7$

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

$- 2 + x - 3 < 0$

Solve the inequality.

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Subtract $3$ from $- 2$ .

$x - 5 < 0$

Add $5$ to both sides of the inequality.

$x < 5$

In the piece where $- 2 + x - 3$ is negative, remove the absolute value and multiply by $- 1$ .

$- (- 2 + x - 3) < 7$

Write as a piecewise.

${\begin{cases} - 2 + x - 3 < 7 & x \geq 5 \\ - (- 2 + x - 3) < 7 & x < 5 \end{cases}$

Subtract $3$ from $- 2$ .

${\begin{cases} x - 5 < 7 & x \geq 5 \\ - (- 2 + x - 3) < 7 & x < 5 \end{cases}$

Simplify $- (- 2 + x - 3) < 7$ .

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Subtract $3$ from $- 2$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 5$ .

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

Solve $x - 5 < 7$ when $x \geq 5$ .

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

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Add $5$ to both sides of the inequality.

$x < 7 + 5$

Add $7$ and $5$ .

$x < 12$

Find the intersection of $x < 12$ and $x \geq 5$ .

$5 \leq x < 12$

Solve $- x + 5 < 7$ when $x < 5$ .

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Solve $- x + 5 < 7$ for $x$ .

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

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Subtract $5$ from both sides of the inequality.

$- x < 7 - 5$

Subtract $5$ from $7$ .

$- x < 2$

Multiply each term in $- x < 2$ by $- 1$

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Multiply each term in $- x < 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$(- x) \cdot - 1 > 2 \cdot - 1$

Multiply $(- x) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 x > 2 \cdot - 1$

Multiply $x$ by $1$ .

$x > 2 \cdot - 1$

Multiply $2$ by $- 1$ .

$x > - 2$

Find the intersection of $x > - 2$ and $x < 5$ .

$- 2 < x < 5$

Find the union of the solutions.

$- 2 < x < 12$

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 12$

Interval Notation:

$(- 2, 12)$

Solve for x |-2+(x-3)|<7

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