Algebra Examples

$| x | > x - 1$

Step 1

Write $| x | > x - 1$ 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.

$x \geq 0$

Step 1.2

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

$x > x - 1$

Step 1.3

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

$x < 0$

Step 1.4

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

$- x > x - 1$

Step 1.5

Write as a piecewise.

${\begin{cases} x > x - 1 & x \geq 0 \\ - x > x - 1 & x < 0 \end{cases}$

Step 2

Solve $x > x - 1$ when $x \geq 0$ .

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Step 2.1

Solve $x > x - 1$ for $x$ .

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Step 2.1.1

Move all terms containing $x$ to the left side of the inequality.

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Step 2.1.1.1

Subtract $x$ from both sides of the inequality.

$x - x > - 1$

Step 2.1.1.2

Subtract $x$ from $x$ .

$0 > - 1$

Step 2.1.2

Since $0 > - 1$ , the inequality will always be true.

Always true

Step 2.2

Find the intersection.

$x \geq 0$

Step 3

Solve $- x > x - 1$ when $x < 0$ .

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Step 3.1

Solve $- x > x - 1$ for $x$ .

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Step 3.1.1

Move all terms containing $x$ to the left side of the inequality.

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Step 3.1.1.1

Subtract $x$ from both sides of the inequality.

$- x - x > - 1$

Step 3.1.1.2

Subtract $x$ from $- x$ .

$- 2 x > - 1$

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

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

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Dividing two negative values results in a positive value.

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

Step 3.2

Find the intersection of $x < \frac{1}{2}$ and $x < 0$ .

$x < 0$

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