Calculus Examples

$| x - 4 | < 2$

Step 1

Write $| x - 4 | < 2$ 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.

$x - 4 \geq 0$

Step 1.2

Add $4$ to both sides of the inequality.

$x \geq 4$

Step 1.3

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

$x - 4 < 2$

Step 1.4

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

$x - 4 < 0$

Step 1.5

Add $4$ to both sides of the inequality.

$x < 4$

Step 1.6

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

$- (x - 4) < 2$

Step 1.7

Write as a piecewise.

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

Step 1.8

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

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Step 1.8.1

Apply the distributive property.

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

Step 1.8.2

Multiply $- 1$ by $- 4$ .

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

Add $4$ to both sides of the inequality.

$x < 2 + 4$

Step 2.1.2

Add $2$ and $4$ .

$x < 6$

Step 2.2

Find the intersection of $x < 6$ and $x \geq 4$ .

$4 \leq x < 6$

Step 3

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

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Step 3.1

Solve $- x + 4 < 2$ 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.

$- x < 2 - 4$

Step 3.1.1.2

Subtract $4$ from $2$ .

$- x < - 2$

Step 3.1.2

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

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Step 3.1.2.1

Divide each term in $- x < - 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

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.

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

Step 3.1.2.2.2

Divide $x$ by $1$ .

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

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $- 2$ by $- 1$ .

$x > 2$

Step 3.2

Find the intersection of $x > 2$ and $x < 4$ .

$2 < x < 4$

Step 4

Find the union of the solutions.

$2 < x < 6$

Step 5

Convert the inequality to interval notation.

$(2, 6)$

Step 6