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

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

Step 1

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

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.

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.

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

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 $- 5$ .

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

Step 2

Solve $x - 5 < 7$ 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.

$x < 7 + 5$

Add $7$ and $5$ .

$x < 12$

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

$5 \leq x < 12$

Step 3

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

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

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

Subtract $5$ from both sides of the inequality.

$- x < 7 - 5$

Subtract $5$ from $7$ .

$- x < 2$

Divide each term in $- x < 2$ by $- 1$ and simplify.

Divide 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.

$\frac{- x}{- 1} > \frac{2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{2}{- 1}$

Divide $x$ by $1$ .

$x > \frac{2}{- 1}$

Simplify the right side.

Divide $2$ by $- 1$ .

$x > - 2$

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

$- 2 < x < 5$

Step 4

Find the union of the solutions.

$- 2 < x < 12$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 2 < x < 12$

Interval Notation:

$(- 2, 12)$

Step 6

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

Related Questions

Support LCMGCF.com by disabling your adblocker.