Algebra Examples

$| x + 3 | > x - 2$

Step 1

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

$x + 3 \geq 0$

Step 1.2

Subtract $3$ from both sides of the inequality.

$x \geq - 3$

Step 1.3

In the piece where $x + 3$ is non-negative, remove the absolute value.

$x + 3 > x - 2$

Step 1.4

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

$x + 3 < 0$

Step 1.5

Subtract $3$ from both sides of the inequality.

$x < - 3$

Step 1.6

In the piece where $x + 3$ is negative, remove the absolute value and multiply by $- 1$ .

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

Step 1.7

Write as a piecewise.

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

Step 1.8

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

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Step 1.8.1

Apply the distributive property.

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

Step 1.8.2

Multiply $- 1$ by $3$ .

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

Step 2

Solve $x + 3 > x - 2$ when $x \geq - 3$ .

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Step 2.1

Solve $x + 3 > x - 2$ 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.

$x + 3 - x > - 2$

Step 2.1.1.2

Combine the opposite terms in $x + 3 - x$ .

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Step 2.1.1.2.1

Subtract $x$ from $x$ .

$0 + 3 > - 2$

Step 2.1.1.2.2

Add $0$ and $3$ .

$3 > - 2$

Step 2.1.2

Since $3 > - 2$ , the inequality will always be true.

Always true

Step 2.2

Find the intersection.

$x \geq - 3$

Step 3

Solve $- x - 3 > x - 2$ when $x < - 3$ .

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Step 3.1

Solve $- x - 3 > x - 2$ 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.

$- x - 3 - x > - 2$

Step 3.1.1.2

Subtract $x$ from $- x$ .

$- 2 x - 3 > - 2$

Step 3.1.2

Move all terms not containing $x$ to the right side of the inequality.

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Step 3.1.2.1

Add $3$ to both sides of the inequality.

$- 2 x > - 2 + 3$

Step 3.1.2.2

Add $- 2$ and $3$ .

$- 2 x > 1$

Step 3.1.3

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

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Step 3.1.3.1

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 3.1.3.2

Simplify the left side.

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Step 3.1.3.2.1

Cancel the common factor of $- 2$ .

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Step 3.1.3.2.1.1

Cancel the common factor.

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

Step 3.1.3.2.1.2

Divide $x$ by $1$ .

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

Step 3.1.3.3

Simplify the right side.

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Step 3.1.3.3.1

Move the negative in front of the fraction.

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

Step 3.2

Find the intersection of $x < - \frac{1}{2}$ and $x < - 3$ .

$x < - 3$

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