Algebra Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

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

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

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

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

Tap for more steps...

Step 1.3.1.1

Factor $2$ out of $2$ .

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

Step 1.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.3.1.2.1

Factor $2$ out of $8$ .

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

Step 1.3.1.2.2

Cancel the common factor.

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

Step 1.3.1.2.3

Rewrite the expression.

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Write as a piecewise.

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

Step 2.8

Apply the distributive property.

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

Step 3

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

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Combine the numerators over the common denominator.

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

Step 3.1.3

Subtract $3$ from $1$ .

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

Step 3.1.4

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

Tap for more steps...

Step 3.1.4.1

Factor $2$ out of $- 2$ .

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

Step 3.1.4.2

Cancel the common factors.

Tap for more steps...

Step 3.1.4.2.1

Factor $2$ out of $4$ .

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

Step 3.1.4.2.2

Cancel the common factor.

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

Step 3.1.4.2.3

Rewrite the expression.

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

Step 3.1.5

Move the negative in front of the fraction.

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

Step 3.2

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

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

Step 4

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

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

Tap for more steps...

Step 4.1.1.1

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

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

Step 4.1.1.2

Combine the numerators over the common denominator.

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

Step 4.1.1.3

Add $1$ and $3$ .

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

Step 4.1.1.4

Divide $4$ by $4$ .

$- x < 1$

Step 4.1.2

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

Tap for more steps...

Step 4.1.2.1

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.

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

Step 4.1.2.2

Simplify the left side.

Tap for more steps...

Step 4.1.2.2.1

Dividing two negative values results in a positive value.

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

Step 4.1.2.2.2

Divide $x$ by $1$ .

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

Step 4.1.2.3

Simplify the right side.

Tap for more steps...

Step 4.1.2.3.1

Divide $1$ by $- 1$ .

$x > - 1$

Step 4.2

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

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

Step 5

Find the union of the solutions.

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

Step 6

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

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

Step 7