Solve the Absolute Value Inequality for x 5|x-3|<35

$5 | x - 3 | < 35$

Divide each term in $5 | x - 3 | < 35$ by $5$ and simplify.

Divide each term in $5 | x - 3 | < 35$ by $5$ .

$\frac{5 | x - 3 |}{5} < \frac{35}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 | x - 3 |}{5} < \frac{35}{5}$

Divide $| x - 3 |$ by $1$ .

$| x - 3 | < \frac{35}{5}$

Simplify the right side.

Divide $35$ by $5$ .

$| x - 3 | < 7$

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

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

$x - 3 \geq 0$

Add $3$ to both sides of the inequality.

$x \geq 3$

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

$x - 3 < 7$

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

$x - 3 < 0$

Add $3$ to both sides of the inequality.

$x < 3$

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

$- (x - 3) < 7$

Write as a piecewise.

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

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

Apply the distributive property.

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

Multiply $- 1$ by $- 3$ .

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

Solve $x - 3 < 7$ when $x \geq 3$ .

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

Add $3$ to both sides of the inequality.

$x < 7 + 3$

Add $7$ and $3$ .

$x < 10$

Find the intersection of $x < 10$ and $x \geq 3$ .

$3 \leq x < 10$

Solve $- x + 3 < 7$ when $x < 3$ .

Solve $- x + 3 < 7$ for $x$ .

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

Subtract $3$ from both sides of the inequality.

$- x < 7 - 3$

Subtract $3$ from $7$ .

$- x < 4$

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 $x < 3$ .

$- 4 < x < 3$

Find the union of the solutions.

$- 4 < x < 10$

The result can be shown in multiple forms.

Inequality Form:

$- 4 < x < 10$

Interval Notation:

$(- 4, 10)$

Solve the Absolute Value Inequality for x 5|x-3|<35

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