Calculus Examples

$| 2 x - 4 | < 6$

Step 1

Write $| 2 x - 4 | < 6$ 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.

$2 x - 4 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Add $4$ to both sides of the inequality.

$2 x \geq 4$

Step 1.2.2

Divide each term in $2 x \geq 4$ by $2$ and simplify.

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Step 1.2.2.1

Divide each term in $2 x \geq 4$ by $2$ .

$\frac{2 x}{2} \geq \frac{4}{2}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{4}{2}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{4}{2}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $4$ by $2$ .

$x \geq 2$

Step 1.3

In the piece where $2 x - 4$ is non-negative, remove the absolute value.

$2 x - 4 < 6$

Step 1.4

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

$2 x - 4 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Add $4$ to both sides of the inequality.

$2 x < 4$

Step 1.5.2

Divide each term in $2 x < 4$ by $2$ and simplify.

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Step 1.5.2.1

Divide each term in $2 x < 4$ by $2$ .

$\frac{2 x}{2} < \frac{4}{2}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $2$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{4}{2}$

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{4}{2}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Divide $4$ by $2$ .

$x < 2$

Step 1.6

In the piece where $2 x - 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x - 4) < 6$

Step 1.7

Write as a piecewise.

${\begin{cases} 2 x - 4 < 6 & x \geq 2 \\ - (2 x - 4) < 6 & x < 2 \end{cases}$

Step 1.8

Simplify $- (2 x - 4) < 6$ .

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Step 1.8.1

Apply the distributive property.

${\begin{cases} 2 x - 4 < 6 & x \geq 2 \\ - (2 x) - - 4 < 6 & x < 2 \end{cases}$

Step 1.8.2

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x - 4 < 6 & x \geq 2 \\ - 2 x - - 4 < 6 & x < 2 \end{cases}$

Step 1.8.3

Multiply $- 1$ by $- 4$ .

${\begin{cases} 2 x - 4 < 6 & x \geq 2 \\ - 2 x + 4 < 6 & x < 2 \end{cases}$

Step 2

Solve $2 x - 4 < 6$ when $x \geq 2$ .

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Step 2.1

Solve $2 x - 4 < 6$ for $x$ .

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Step 2.1.1

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

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Step 2.1.1.1

Add $4$ to both sides of the inequality.

$2 x < 6 + 4$

Step 2.1.1.2

Add $6$ and $4$ .

$2 x < 10$

Step 2.1.2

Divide each term in $2 x < 10$ by $2$ and simplify.

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Step 2.1.2.1

Divide each term in $2 x < 10$ by $2$ .

$\frac{2 x}{2} < \frac{10}{2}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} < \frac{10}{2}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{10}{2}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Divide $10$ by $2$ .

$x < 5$

Step 2.2

Find the intersection of $x < 5$ and $x \geq 2$ .

$2 \leq x < 5$

Step 3

Solve $- 2 x + 4 < 6$ when $x < 2$ .

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Step 3.1

Solve $- 2 x + 4 < 6$ for $x$ .

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Step 3.1.1

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

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Step 3.1.1.1

Subtract $4$ from both sides of the inequality.

$- 2 x < 6 - 4$

Step 3.1.1.2

Subtract $4$ from $6$ .

$- 2 x < 2$

Step 3.1.2

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

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Step 3.1.2.1

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

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

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.

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

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

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

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $2$ by $- 2$ .

$x > - 1$

Step 3.2

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

$- 1 < x < 2$

Step 4

Find the union of the solutions.

$- 1 < x < 5$

Step 5

Convert the inequality to interval notation.

$(- 1, 5)$

Step 6