Pre-Algebra Examples

$| 2 x - 9 | - 3 > 4$

Step 1

Write $| 2 x - 9 | - 3 > 4$ 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.

$2 x - 9 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Add $9$ to both sides of the inequality.

$2 x \geq 9$

Step 1.2.2

Divide each term in $2 x \geq 9$ by $2$ and simplify.

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Step 1.2.2.1

Divide each term in $2 x \geq 9$ by $2$ .

$\frac{2 x}{2} \geq \frac{9}{2}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{9}{2}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{9}{2}$

Step 1.3

In the piece where $2 x - 9$ is non-negative, remove the absolute value.

$2 x - 9 - 3 > 4$

Step 1.4

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

$2 x - 9 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Add $9$ to both sides of the inequality.

$2 x < 9$

Step 1.5.2

Divide each term in $2 x < 9$ by $2$ and simplify.

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Step 1.5.2.1

Divide each term in $2 x < 9$ by $2$ .

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

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $2$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

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

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.6

In the piece where $2 x - 9$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x - 9) - 3 > 4$

Step 1.7

Write as a piecewise.

${\begin{cases} 2 x - 9 - 3 > 4 & x \geq \frac{9}{2} \\ - (2 x - 9) - 3 > 4 & x < \frac{9}{2} \end{cases}$

Step 1.8

Subtract $3$ from $- 9$ .

${\begin{cases} 2 x - 12 > 4 & x \geq \frac{9}{2} \\ - (2 x - 9) - 3 > 4 & x < \frac{9}{2} \end{cases}$

Step 1.9

Simplify $- (2 x - 9) - 3 > 4$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 2 x - 12 > 4 & x \geq \frac{9}{2} \\ - (2 x) - - 9 - 3 > 4 & x < \frac{9}{2} \end{cases}$

Step 1.9.1.2

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x - 12 > 4 & x \geq \frac{9}{2} \\ - 2 x - - 9 - 3 > 4 & x < \frac{9}{2} \end{cases}$

Step 1.9.1.3

Multiply $- 1$ by $- 9$ .

${\begin{cases} 2 x - 12 > 4 & x \geq \frac{9}{2} \\ - 2 x + 9 - 3 > 4 & x < \frac{9}{2} \end{cases}$

Step 1.9.2

Subtract $3$ from $9$ .

${\begin{cases} 2 x - 12 > 4 & x \geq \frac{9}{2} \\ - 2 x + 6 > 4 & x < \frac{9}{2} \end{cases}$

Step 2

Solve $2 x - 12 > 4$ for $x$ .

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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 $12$ to both sides of the inequality.

$2 x > 4 + 12$

Step 2.1.2

Add $4$ and $12$ .

$2 x > 16$

Step 2.2

Divide each term in $2 x > 16$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x > 16$ by $2$ .

$\frac{2 x}{2} > \frac{16}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} > \frac{16}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{16}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $16$ by $2$ .

$x > 8$

Step 3

Solve $- 2 x + 6 > 4$ for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $6$ from both sides of the inequality.

$- 2 x > 4 - 6$

Step 3.1.2

Subtract $6$ from $4$ .

$- 2 x > - 2$

Step 3.2

Divide each term in $- 2 x > - 2$ by $- 2$ and simplify.

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Step 3.2.1

Divide each term in $- 2 x > - 2$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $- 2$ by $- 2$ .

$x < 1$

Step 4

Find the union of the solutions.

$x < 1$ or $x > 8$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < 1 or x > 8$

Interval Notation:

$(- \infty, 1) \cup (8, \infty)$

Step 6