Algebra Examples

$| x - \frac{1}{2} | < 5$

Step 1

Write $| x - \frac{1}{2} | < 5$ as a piecewise.

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Step 1.1

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

$x - \frac{1}{2} \geq 0$

Step 1.2

Add $\frac{1}{2}$ to both sides of the inequality.

$x \geq \frac{1}{2}$

Step 1.3

In the piece where $x - \frac{1}{2}$ is non-negative, remove the absolute value.

$x - \frac{1}{2} < 5$

Step 1.4

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

$x - \frac{1}{2} < 0$

Step 1.5

Add $\frac{1}{2}$ to both sides of the inequality.

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

Step 1.6

In the piece where $x - \frac{1}{2}$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - \frac{1}{2}) < 5$

Step 1.7

Write as a piecewise.

${\begin{cases} x - \frac{1}{2} < 5 & x \geq \frac{1}{2} \\ - (x - \frac{1}{2}) < 5 & x < \frac{1}{2} \end{cases}$

Step 1.8

Simplify $- (x - \frac{1}{2}) < 5$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} x - \frac{1}{2} < 5 & x \geq \frac{1}{2} \\ - x - - \frac{1}{2} < 5 & x < \frac{1}{2} \end{cases}$

Step 1.8.2

Multiply $- - \frac{1}{2}$ .

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Step 1.8.2.1

Multiply $- 1$ by $- 1$ .

${\begin{cases} x - \frac{1}{2} < 5 & x \geq \frac{1}{2} \\ - x + 1 (\frac{1}{2}) < 5 & x < \frac{1}{2} \end{cases}$

Step 1.8.2.2

Multiply $\frac{1}{2}$ by $1$ .

${\begin{cases} x - \frac{1}{2} < 5 & x \geq \frac{1}{2} \\ - x + \frac{1}{2} < 5 & x < \frac{1}{2} \end{cases}$

Step 2

Solve $x - \frac{1}{2} < 5$ when $x \geq \frac{1}{2}$ .

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Step 2.1

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

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Step 2.1.1

Add $\frac{1}{2}$ to both sides of the inequality.

$x < 5 + \frac{1}{2}$

Step 2.1.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x < 5 \cdot \frac{2}{2} + \frac{1}{2}$

Step 2.1.3

Combine $5$ and $\frac{2}{2}$ .

$x < \frac{5 \cdot 2}{2} + \frac{1}{2}$

Step 2.1.4

Combine the numerators over the common denominator.

$x < \frac{5 \cdot 2 + 1}{2}$

Step 2.1.5

Simplify the numerator.

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Step 2.1.5.1

Multiply $5$ by $2$ .

$x < \frac{10 + 1}{2}$

Step 2.1.5.2

Add $10$ and $1$ .

$x < \frac{11}{2}$

Step 2.2

Find the intersection of $x < \frac{11}{2}$ and $x \geq \frac{1}{2}$ .

$\frac{1}{2} \leq x < \frac{11}{2}$

Step 3

Solve $- x + \frac{1}{2} < 5$ when $x < \frac{1}{2}$ .

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Step 3.1

Solve $- x + \frac{1}{2} < 5$ for $x$ .

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Step 3.1.1

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

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Step 3.1.1.1

Subtract $\frac{1}{2}$ from both sides of the inequality.

$- x < 5 - \frac{1}{2}$

Step 3.1.1.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- x < 5 \cdot \frac{2}{2} - \frac{1}{2}$

Step 3.1.1.3

Combine $5$ and $\frac{2}{2}$ .

$- x < \frac{5 \cdot 2}{2} - \frac{1}{2}$

Step 3.1.1.4

Combine the numerators over the common denominator.

$- x < \frac{5 \cdot 2 - 1}{2}$

Step 3.1.1.5

Simplify the numerator.

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Step 3.1.1.5.1

Multiply $5$ by $2$ .

$- x < \frac{10 - 1}{2}$

Step 3.1.1.5.2

Subtract $1$ from $10$ .

$- x < \frac{9}{2}$

Step 3.1.2

Divide each term in $- x < \frac{9}{2}$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- x < \frac{9}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{\frac{9}{2}}{- 1}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{\frac{9}{2}}{- 1}$

Step 3.1.2.2.2

Divide $x$ by $1$ .

$x > \frac{\frac{9}{2}}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Move the negative one from the denominator of $\frac{\frac{9}{2}}{- 1}$ .

$x > - 1 \cdot \frac{9}{2}$

Step 3.1.2.3.2

Rewrite $- 1 \cdot \frac{9}{2}$ as $- \frac{9}{2}$ .

$x > - \frac{9}{2}$

Step 3.2

Find the intersection of $x > - \frac{9}{2}$ and $x < \frac{1}{2}$ .

$- \frac{9}{2} < x < \frac{1}{2}$

Step 4

Find the union of the solutions.

$- \frac{9}{2} < x < \frac{11}{2}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- \frac{9}{2} < x < \frac{11}{2}$

Interval Notation:

$(- \frac{9}{2}, \frac{11}{2})$

Step 6