Algebra Examples

$| x - 2 | + 3 < 0$

Step 1

Write $| x - 2 | + 3 < 0$ 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.

$x - 2 + 3 < 0$

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 $- 1$ .

$- (x - 2) + 3 < 0$

Step 1.7

Write as a piecewise.

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

Step 1.8

Add $- 2$ and $3$ .

${\begin{cases} x + 1 < 0 & x \geq 2 \\ - (x - 2) + 3 < 0 & x < 2 \end{cases}$

Step 1.9

Simplify $- (x - 2) + 3 < 0$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} x + 1 < 0 & x \geq 2 \\ - x - - 2 + 3 < 0 & x < 2 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $- 2$ .

${\begin{cases} x + 1 < 0 & x \geq 2 \\ - x + 2 + 3 < 0 & x < 2 \end{cases}$

Step 1.9.2

Add $2$ and $3$ .

${\begin{cases} x + 1 < 0 & x \geq 2 \\ - x + 5 < 0 & x < 2 \end{cases}$

Step 2

Solve $x + 1 < 0$ when $x \geq 2$ .

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Step 2.1

Subtract $1$ from both sides of the inequality.

$x < - 1$

Step 2.2

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

No solution

Step 3

Solve $- x + 5 < 0$ when $x < 2$ .

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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 $x < 2$ .

No solution

Step 4

Find the union of the solutions.

No solution