Solve the Inequality for x 8|x+3/4|<2

$8 | x + \frac{3}{4} | < 2$

Divide each term in $8 | x + \frac{3}{4} | < 2$ by $8$ and simplify.

Divide each term in $8 | x + \frac{3}{4} | < 2$ by $8$ .

$\frac{8 | x + \frac{3}{4} |}{8} < \frac{2}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 | x + \frac{3}{4} |}{8} < \frac{2}{8}$

Divide $| x + \frac{3}{4} |$ by $1$ .

$| x + \frac{3}{4} | < \frac{2}{8}$

Simplify the right side.

Cancel the common factor of $2$ and $8$ .

Factor $2$ out of $2$ .

$| x + \frac{3}{4} | < \frac{2 (1)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$| x + \frac{3}{4} | < \frac{2 \cdot 1}{2 \cdot 4}$

Cancel the common factor.

$| x + \frac{3}{4} | < \frac{2 \cdot 1}{2 \cdot 4}$

Rewrite the expression.

$| x + \frac{3}{4} | < \frac{1}{4}$

Write $| x + \frac{3}{4} | < \frac{1}{4}$ as a piecewise.

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

$x + \frac{3}{4} \geq 0$

Subtract $\frac{3}{4}$ from both sides of the inequality.

$x \geq - \frac{3}{4}$

In the piece where $x + \frac{3}{4}$ is non-negative, remove the absolute value.

$x + \frac{3}{4} < \frac{1}{4}$

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

$x + \frac{3}{4} < 0$

Subtract $\frac{3}{4}$ from both sides of the inequality.

$x < - \frac{3}{4}$

In the piece where $x + \frac{3}{4}$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + \frac{3}{4}) < \frac{1}{4}$

Write as a piecewise.

${\begin{cases} x + \frac{3}{4} < \frac{1}{4} & x \geq - \frac{3}{4} \\ - (x + \frac{3}{4}) < \frac{1}{4} & x < - \frac{3}{4} \end{cases}$

Apply the distributive property.

${\begin{cases} x + \frac{3}{4} < \frac{1}{4} & x \geq - \frac{3}{4} \\ - x - \frac{3}{4} < \frac{1}{4} & x < - \frac{3}{4} \end{cases}$

Solve $x + \frac{3}{4} < \frac{1}{4}$ when $x \geq - \frac{3}{4}$ .

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

Subtract $\frac{3}{4}$ from both sides of the inequality.

$x < \frac{1}{4} - \frac{3}{4}$

Combine the numerators over the common denominator.

$x < \frac{1 - 3}{4}$

Subtract $3$ from $1$ .

$x < \frac{- 2}{4}$

Cancel the common factor of $- 2$ and $4$ .

Factor $2$ out of $- 2$ .

$x < \frac{2 (- 1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x < \frac{2 \cdot - 1}{2 \cdot 2}$

Cancel the common factor.

$x < \frac{2 \cdot - 1}{2 \cdot 2}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Find the intersection of $x < - \frac{1}{2}$ and $x \geq - \frac{3}{4}$ .

$- \frac{3}{4} \leq x < - \frac{1}{2}$

Solve $- x - \frac{3}{4} < \frac{1}{4}$ when $x < - \frac{3}{4}$ .

Solve $- x - \frac{3}{4} < \frac{1}{4}$ for $x$ .

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

Add $\frac{3}{4}$ to both sides of the inequality.

$- x < \frac{1}{4} + \frac{3}{4}$

Combine the numerators over the common denominator.

$- x < \frac{1 + 3}{4}$

Add $1$ and $3$ .

$- x < \frac{4}{4}$

Divide $4$ by $4$ .

$- x < 1$

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 $x < - \frac{3}{4}$ .

$- 1 < x < - \frac{3}{4}$

Find the union of the solutions.

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

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(- 1, - \frac{1}{2})$

Solve the Inequality for x 8|x+3/4|<2

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