Pre-Algebra Examples

| x | + 7 > 9

$| x | + 7 > 9$

Step 1

Write

| x | + 7 > 9

$| x | + 7 > 9$ 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 \geq 0

$x \geq 0$

Step 1.2

In the piece where

x

$x$ is non-negative, remove the absolute value.

x + 7 > 9

$x + 7 > 9$

Step 1.3

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

x < 0

$x < 0$

Step 1.4

In the piece where

x

$x$ is negative, remove the absolute value and multiply by

- 1

$- 1$ .

- x + 7 > 9

$- x + 7 > 9$

Step 1.5

Write as a piecewise.

{\begin{matrix} x + 7 > 9 & x \geq 0 - x + 7 > 9 & x < 0 \end{matrix}

${\begin{cases} x + 7 > 9 & x \geq 0 \\ - x + 7 > 9 & x < 0 \end{cases}$

{\begin{matrix} x + 7 > 9 & x \geq 0 - x + 7 > 9 & x < 0 \end{matrix}

${\begin{cases} x + 7 > 9 & x \geq 0 \\ - x + 7 > 9 & x < 0 \end{cases}$

Step 2

Move all terms not containing

x

$x$ to the right side of the inequality.

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Step 2.1

Subtract

7

$7$ from both sides of the inequality.

x > 9 - 7

$x > 9 - 7$

Step 2.2

Subtract

7

$7$ from

9

$9$ .

x > 2

$x > 2$

x > 2

$x > 2$

Step 3

Solve

- x + 7 > 9

$- x + 7 > 9$ for

x

$x$ .

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Step 3.1

Move all terms not containing

x

$x$ to the right side of the inequality.

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Step 3.1.1

Subtract

7

$7$ from both sides of the inequality.

- x > 9 - 7

$- x > 9 - 7$

Step 3.1.2

Subtract

7

$7$ from

9

$9$ .

- x > 2

$- x > 2$

- x > 2

$- x > 2$

Step 3.2

Divide each term in

- x > 2

$- x > 2$ by

- 1

$- 1$ and simplify.

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Step 3.2.1

Divide each term in

- x > 2

$- x > 2$ by

- 1

$- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

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

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{2}{- 1}

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

Step 3.2.2.2

Divide

x

$x$ by

1

$1$ .

x < \frac{2}{- 1}

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

x < \frac{2}{- 1}

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide

2

$2$ by

- 1

$- 1$ .

x < - 2

$x < - 2$

x < - 2

$x < - 2$

x < - 2

$x < - 2$

x < - 2

$x < - 2$

Step 4

Find the union of the solutions.

x < - 2

$x < - 2$ or

x > 2

$x > 2$

Step 5

The result can be shown in multiple forms.

Inequality Form:

x < - 2 or x > 2

$x < - 2 or x > 2$

Interval Notation:

(- \infty, - 2) \cup (2, \infty)

$(- \infty, - 2) \cup (2, \infty)$

Step 6