Finite Math Examples

$3 | x - 2 | + 12 > 0$

Step 1

Write $3 | x - 2 | + 12 > 0$ 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.

$x - 2 \geq 0$

Step 1.2

Add $2$ to both sides of the inequality.

$x \geq 2$

Step 1.3

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

$3 (x - 2) + 12 > 0$

Step 1.4

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

$x - 2 < 0$

Step 1.5

Add $2$ to both sides of the inequality.

$x < 2$

Step 1.6

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

$3 (- (x - 2)) + 12 > 0$

Step 1.7

Write as a piecewise.

${\begin{cases} 3 (x - 2) + 12 > 0 & x \geq 2 \\ 3 (- (x - 2)) + 12 > 0 & x < 2 \end{cases}$

Step 1.8

Simplify $3 (x - 2) + 12 > 0$ .

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Step 1.8.1

Simplify each term.

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Step 1.8.1.1

Apply the distributive property.

${\begin{cases} 3 x + 3 \cdot - 2 + 12 > 0 & x \geq 2 \\ 3 (- (x - 2)) + 12 > 0 & x < 2 \end{cases}$

Step 1.8.1.2

Multiply $3$ by $- 2$ .

${\begin{cases} 3 x - 6 + 12 > 0 & x \geq 2 \\ 3 (- (x - 2)) + 12 > 0 & x < 2 \end{cases}$

Step 1.8.2

Add $- 6$ and $12$ .

${\begin{cases} 3 x + 6 > 0 & x \geq 2 \\ 3 (- (x - 2)) + 12 > 0 & x < 2 \end{cases}$

Step 1.9

Simplify $3 (- (x - 2)) + 12 > 0$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 3 x + 6 > 0 & x \geq 2 \\ 3 (- x - - 2) + 12 > 0 & x < 2 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $- 2$ .

${\begin{cases} 3 x + 6 > 0 & x \geq 2 \\ 3 (- x + 2) + 12 > 0 & x < 2 \end{cases}$

Step 1.9.1.3

Apply the distributive property.

${\begin{cases} 3 x + 6 > 0 & x \geq 2 \\ 3 (- x) + 3 \cdot 2 + 12 > 0 & x < 2 \end{cases}$

Step 1.9.1.4

Multiply $- 1$ by $3$ .

${\begin{cases} 3 x + 6 > 0 & x \geq 2 \\ - 3 x + 3 \cdot 2 + 12 > 0 & x < 2 \end{cases}$

Step 1.9.1.5

Multiply $3$ by $2$ .

${\begin{cases} 3 x + 6 > 0 & x \geq 2 \\ - 3 x + 6 + 12 > 0 & x < 2 \end{cases}$

Step 1.9.2

Add $6$ and $12$ .

${\begin{cases} 3 x + 6 > 0 & x \geq 2 \\ - 3 x + 18 > 0 & x < 2 \end{cases}$

Step 2

Solve $3 x + 6 > 0$ when $x \geq 2$ .

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Step 2.1

Solve $3 x + 6 > 0$ for $x$ .

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Step 2.1.1

Subtract $6$ from both sides of the inequality.

$3 x > - 6$

Step 2.1.2

Divide each term in $3 x > - 6$ by $3$ and simplify.

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Step 2.1.2.1

Divide each term in $3 x > - 6$ by $3$ .

$\frac{3 x}{3} > \frac{- 6}{3}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $3$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} > \frac{- 6}{3}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 6}{3}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Divide $- 6$ by $3$ .

$x > - 2$

Step 2.2

Find the intersection of $x > - 2$ and $x \geq 2$ .

$x \geq 2$

Step 3

Solve $- 3 x + 18 > 0$ when $x < 2$ .

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Step 3.1

Solve $- 3 x + 18 > 0$ for $x$ .

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Step 3.1.1

Subtract $18$ from both sides of the inequality.

$- 3 x > - 18$

Step 3.1.2

Divide each term in $- 3 x > - 18$ by $- 3$ and simplify.

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Step 3.1.2.1

Divide each term in $- 3 x > - 18$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} < \frac{- 18}{- 3}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} < \frac{- 18}{- 3}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 18}{- 3}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $- 18$ by $- 3$ .

$x < 6$

Step 3.2

Find the intersection of $x < 6$ and $x < 2$ .

$x < 2$

Step 4

Find the union of the solutions.

All real numbers

Step 5

Convert the inequality to interval notation.

$(- \infty, \infty)$

Step 6