Pre-Algebra Examples

$| x + 1 | + 13 > 10$

Step 1

Write $| x + 1 | + 13 > 10$ 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.

$x + 1 \geq 0$

Step 1.2

Subtract $1$ from both sides of the inequality.

$x \geq - 1$

Step 1.3

In the piece where $x + 1$ is non-negative, remove the absolute value.

$x + 1 + 13 > 10$

Step 1.4

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

$x + 1 < 0$

Step 1.5

Subtract $1$ from both sides of the inequality.

$x < - 1$

Step 1.6

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

$- (x + 1) + 13 > 10$

Step 1.7

Write as a piecewise.

${\begin{cases} x + 1 + 13 > 10 & x \geq - 1 \\ - (x + 1) + 13 > 10 & x < - 1 \end{cases}$

Step 1.8

Add $1$ and $13$ .

${\begin{cases} x + 14 > 10 & x \geq - 1 \\ - (x + 1) + 13 > 10 & x < - 1 \end{cases}$

Step 1.9

Simplify $- (x + 1) + 13 > 10$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} x + 14 > 10 & x \geq - 1 \\ - x - 1 \cdot 1 + 13 > 10 & x < - 1 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $1$ .

${\begin{cases} x + 14 > 10 & x \geq - 1 \\ - x - 1 + 13 > 10 & x < - 1 \end{cases}$

Step 1.9.2

Add $- 1$ and $13$ .

${\begin{cases} x + 14 > 10 & x \geq - 1 \\ - x + 12 > 10 & x < - 1 \end{cases}$

Step 2

Solve $x + 14 > 10$ when $x \geq - 1$ .

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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

Subtract $14$ from both sides of the inequality.

$x > 10 - 14$

Step 2.1.2

Subtract $14$ from $10$ .

$x > - 4$

Step 2.2

Find the intersection of $x > - 4$ and $x \geq - 1$ .

$x \geq - 1$

Step 3

Solve $- x + 12 > 10$ when $x < - 1$ .

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Step 3.1

Solve $- x + 12 > 10$ 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 $12$ from both sides of the inequality.

$- x > 10 - 12$

Step 3.1.1.2

Subtract $12$ from $10$ .

$- x > - 2$

Step 3.1.2

Divide each term in $- x > - 2$ by $- 1$ and simplify.

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Step 3.1.2.1

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

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

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.

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

Step 3.1.2.2.2

Divide $x$ by $1$ .

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

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $- 2$ by $- 1$ .

$x < 2$

Step 3.2

Find the intersection of $x < 2$ and $x < - 1$ .

$x < - 1$

Step 4

Find the union of the solutions for any value of $x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Step 6