Algebra Examples

$| x - 5 | < 0$

Step 1

Write $| x - 5 | < 0$ 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.

$x - 5 \geq 0$

Step 1.2

Add $5$ to both sides of the inequality.

$x \geq 5$

Step 1.3

In the piece where $x - 5$ is non-negative, remove the absolute value.

$x - 5 < 0$

Step 1.4

Add $5$ to both sides of the inequality.

$x < 5$

Step 1.5

In the piece where $x - 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 5) < 0$

Step 1.6

Write as a piecewise.

${\begin{cases} x - 5 < 0 & x \geq 5 \\ - (x - 5) < 0 & x < 5 \end{cases}$

Step 1.7

Simplify $- (x - 5) < 0$ .

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Step 1.7.1

Apply the distributive property.

${\begin{cases} x - 5 < 0 & x \geq 5 \\ - x - - 5 < 0 & x < 5 \end{cases}$

Step 1.7.2

Multiply $- 1$ by $- 5$ .

${\begin{cases} x - 5 < 0 & x \geq 5 \\ - x + 5 < 0 & x < 5 \end{cases}$

Step 2

Solve $x - 5 < 0$ when $x \geq 5$ .

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Step 2.1

Add $5$ to both sides of the inequality.

$x < 5$

Step 2.2

Find the intersection of $x < 5$ and $x \geq 5$ .

No solution

Step 3

Solve $- x + 5 < 0$ when $x < 5$ .

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Step 3.1

Solve $- x + 5 < 0$ for $x$ .

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Step 3.1.1

Subtract $5$ from both sides of the inequality.

$- x < - 5$

Step 3.1.2

Divide each term in $- x < - 5$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- x < - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{- 5}{- 1}$

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.

$\frac{x}{1} > \frac{- 5}{- 1}$

Step 3.1.2.2.2

Divide $x$ by $1$ .

$x > \frac{- 5}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $- 5$ by $- 1$ .

$x > 5$

Step 3.2

Find the intersection of $x > 5$ and $x < 5$ .

No solution

Step 4

Find the union of the solutions.

No solution