Algebra Examples

$3 | x + 2 | > x + 5$

Step 1

Write $3 | x + 2 | > x + 5$ 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

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 1.3

In the piece where $x + 2$ is non-negative, remove the absolute value.

$3 (x + 2) > x + 5$

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

Subtract $2$ from 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 $- 1$ .

$3 (- (x + 2)) > x + 5$

Step 1.7

Write as a piecewise.

${\begin{cases} 3 (x + 2) > x + 5 & x \geq - 2 \\ 3 (- (x + 2)) > x + 5 & x < - 2 \end{cases}$

Step 1.8

Simplify $3 (x + 2) > x + 5$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} 3 x + 3 \cdot 2 > x + 5 & x \geq - 2 \\ 3 (- (x + 2)) > x + 5 & x < - 2 \end{cases}$

Step 1.8.2

Multiply $3$ by $2$ .

${\begin{cases} 3 x + 6 > x + 5 & x \geq - 2 \\ 3 (- (x + 2)) > x + 5 & x < - 2 \end{cases}$

Step 1.9

Simplify $3 (- (x + 2)) > x + 5$ .

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Step 1.9.1

Apply the distributive property.

${\begin{cases} 3 x + 6 > x + 5 & x \geq - 2 \\ 3 (- x - 1 \cdot 2) > x + 5 & x < - 2 \end{cases}$

Step 1.9.2

Multiply $- 1$ by $2$ .

${\begin{cases} 3 x + 6 > x + 5 & x \geq - 2 \\ 3 (- x - 2) > x + 5 & x < - 2 \end{cases}$

Step 1.9.3

Apply the distributive property.

${\begin{cases} 3 x + 6 > x + 5 & x \geq - 2 \\ 3 (- x) + 3 \cdot - 2 > x + 5 & x < - 2 \end{cases}$

Step 1.9.4

Multiply $- 1$ by $3$ .

${\begin{cases} 3 x + 6 > x + 5 & x \geq - 2 \\ - 3 x + 3 \cdot - 2 > x + 5 & x < - 2 \end{cases}$

Step 1.9.5

Multiply $3$ by $- 2$ .

${\begin{cases} 3 x + 6 > x + 5 & x \geq - 2 \\ - 3 x - 6 > x + 5 & x < - 2 \end{cases}$

Step 2

Solve $3 x + 6 > x + 5$ for $x$ .

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Step 2.1

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

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Step 2.1.1

Subtract $x$ from both sides of the inequality.

$3 x + 6 - x > 5$

Step 2.1.2

Subtract $x$ from $3 x$ .

$2 x + 6 > 5$

Step 2.2

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

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Step 2.2.1

Subtract $6$ from both sides of the inequality.

$2 x > 5 - 6$

Step 2.2.2

Subtract $6$ from $5$ .

$2 x > - 1$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

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

Step 3

Solve $- 3 x - 6 > x + 5$ for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $x$ from both sides of the inequality.

$- 3 x - 6 - x > 5$

Step 3.1.2

Subtract $x$ from $- 3 x$ .

$- 4 x - 6 > 5$

Step 3.2

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

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Step 3.2.1

Add $6$ to both sides of the inequality.

$- 4 x > 5 + 6$

Step 3.2.2

Add $5$ and $6$ .

$- 4 x > 11$

Step 3.3

Divide each term in $- 4 x > 11$ by $- 4$ and simplify.

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Step 3.3.1

Divide each term in $- 4 x > 11$ by $- 4$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 4 x}{- 4} < \frac{11}{- 4}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 4$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} < \frac{11}{- 4}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x < \frac{11}{- 4}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$x < - \frac{11}{4}$

Step 4

Find the union of the solutions.

$x < - \frac{11}{4}$ or $x > - \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{11}{4} or x > - \frac{1}{2}$

Interval Notation:

$(- \infty, - \frac{11}{4}) \cup (- \frac{1}{2}, \infty)$

Step 6