Algebra Examples

$| x - 3 | > 0$

Step 1

Write $| x - 3 | > 0$ 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.

$x - 3 \geq 0$

Step 1.2

Add $3$ to both sides of the inequality.

$x \geq 3$

Step 1.3

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

$x - 3 > 0$

Step 1.4

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

$x - 3 < 0$

Step 1.5

Add $3$ to both sides of the inequality.

$x < 3$

Step 1.6

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

$- (x - 3) > 0$

Step 1.7

Write as a piecewise.

${\begin{cases} x - 3 > 0 & x \geq 3 \\ - (x - 3) > 0 & x < 3 \end{cases}$

Step 1.8

Simplify $- (x - 3) > 0$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} x - 3 > 0 & x \geq 3 \\ - x - - 3 > 0 & x < 3 \end{cases}$

Step 1.8.2

Multiply $- 1$ by $- 3$ .

${\begin{cases} x - 3 > 0 & x \geq 3 \\ - x + 3 > 0 & x < 3 \end{cases}$

Step 2

Add $3$ to both sides of the inequality.

$x > 3$

Step 3

Solve $- x + 3 > 0$ for $x$ .

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Step 3.1

Subtract $3$ from both sides of the inequality.

$- x > - 3$

Step 3.2

Divide each term in $- x > - 3$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- x > - 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2.2

Divide $x$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 3$ by $- 1$ .

$x < 3$

Step 4

Find the union of the solutions.

$x < 3$ or $x > 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < 3 or x > 3$

Interval Notation:

$(- \infty, 3) \cup (3, \infty)$

Step 6