Algebra Examples

$f (x) < - | x - 4 |$

Step 1

Subtract $f (x)$ from both sides of the inequality.

$0 < - | x - 4 | - f (x)$

Step 2

Find the slope and the y-intercept for the boundary line.

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.1.2

Rewrite so $x$ is on the left side of the inequality.

$- | x - 4 | - f (x) > 0$

Step 2.1.3

Write $- | x - 4 | - f (x) > 0$ as a piecewise.

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Step 2.1.3.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x - 4 \geq 0$

Step 2.1.3.2

Add $4$ to both sides of the inequality.

$x \geq 4$

Step 2.1.3.3

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

$- (x - 4) - f (x) > 0$

Step 2.1.3.4

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

$x - 4 < 0$

Step 2.1.3.5

Add $4$ to both sides of the inequality.

$x < 4$

Step 2.1.3.6

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

$- - (x - 4) - f (x) > 0$

Step 2.1.3.7

Write as a piecewise.

${\begin{cases} - (x - 4) - f (x) > 0 & x \geq 4 \\ - - (x - 4) - f (x) > 0 & x < 4 \end{cases}$

Step 2.1.3.8

Simplify each term.

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Step 2.1.3.8.1

Apply the distributive property.

${\begin{cases} - x - - 4 - f (x) > 0 & x \geq 4 \\ - - (x - 4) - f (x) > 0 & x < 4 \end{cases}$

Step 2.1.3.8.2

Multiply $- 1$ by $- 4$ .

${\begin{cases} - x + 4 - f (x) > 0 & x \geq 4 \\ - - (x - 4) - f (x) > 0 & x < 4 \end{cases}$

Step 2.1.3.9

Simplify each term.

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Step 2.1.3.9.1

Apply the distributive property.

${\begin{cases} - x + 4 - f (x) > 0 & x \geq 4 \\ - (- x - - 4) - f (x) > 0 & x < 4 \end{cases}$

Step 2.1.3.9.2

Multiply $- 1$ by $- 4$ .

${\begin{cases} - x + 4 - f (x) > 0 & x \geq 4 \\ - (- x + 4) - f (x) > 0 & x < 4 \end{cases}$

Step 2.1.3.9.3

Apply the distributive property.

${\begin{cases} - x + 4 - f (x) > 0 & x \geq 4 \\ - - x - 1 \cdot 4 - f (x) > 0 & x < 4 \end{cases}$

Step 2.1.3.9.4

Multiply $- - x$ .

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Step 2.1.3.9.4.1

Multiply $- 1$ by $- 1$ .

${\begin{cases} - x + 4 - f (x) > 0 & x \geq 4 \\ 1 x - 1 \cdot 4 - f (x) > 0 & x < 4 \end{cases}$

Step 2.1.3.9.4.2

Multiply $x$ by $1$ .

${\begin{cases} - x + 4 - f (x) > 0 & x \geq 4 \\ x - 1 \cdot 4 - f (x) > 0 & x < 4 \end{cases}$

Step 2.1.3.9.5

Multiply $- 1$ by $4$ .

${\begin{cases} - x + 4 - f (x) > 0 & x \geq 4 \\ x - 4 - f (x) > 0 & x < 4 \end{cases}$

Step 2.1.4

Solve $- x + 4 - f (x) > 0$ when $x \geq 4$ .

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Step 2.1.4.1

Solve $- x + 4 - f (x) > 0$ for $x$ .

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Step 2.1.4.1.1

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

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Step 2.1.4.1.1.1

Subtract $4$ from both sides of the inequality.

$- x - f (x) > - 4$

Step 2.1.4.1.1.2

Add $f (x)$ to both sides of the inequality.

$- x > - 4 + f (x)$

Step 2.1.4.1.2

Divide each term in $- x > - 4 + f (x)$ by $- 1$ and simplify.

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Step 2.1.4.1.2.1

Divide each term in $- x > - 4 + f (x)$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{- 4}{- 1} + \frac{f (x)}{- 1}$

Step 2.1.4.1.2.2

Simplify the left side.

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Step 2.1.4.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{- 4}{- 1} + \frac{f (x)}{- 1}$

Step 2.1.4.1.2.2.2

Divide $x$ by $1$ .

$x < \frac{- 4}{- 1} + \frac{f (x)}{- 1}$

Step 2.1.4.1.2.3

Simplify the right side.

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Step 2.1.4.1.2.3.1

Simplify each term.

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Step 2.1.4.1.2.3.1.1

Divide $- 4$ by $- 1$ .

$x < 4 + \frac{f (x)}{- 1}$

Step 2.1.4.1.2.3.1.2

Move the negative one from the denominator of $\frac{f (x)}{- 1}$ .

$x < 4 - 1 \cdot f (x)$

Step 2.1.4.1.2.3.1.3

Rewrite $- 1 \cdot f (x)$ as $- f (x)$ .

$x < 4 - f (x)$

Step 2.1.4.2

Find the intersection of $x < 4 - f (x)$ and $x \geq 4$ .

$4 \leq x < 4 - f (x)$

Step 2.1.5

Solve $x - 4 - f (x) > 0$ when $x < 4$ .

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Step 2.1.5.1

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

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Step 2.1.5.1.1

Add $4$ to both sides of the inequality.

$x - f (x) > 4$

Step 2.1.5.1.2

Add $f (x)$ to both sides of the inequality.

$x > 4 + f (x)$

Step 2.1.5.2

Find the intersection of $x > 4 + f (x)$ and $x < 4$ .

$4 + f (x) < x < 4$

Step 2.1.6

Find the union of the solutions.

$N o (Min i m u m) < x < N o (Max i m u m)$

Step 2.2

The equation is not linear, so a constant slope does not exist.

Not Linear

Step 3

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $- | x - 4 | - f (x)$ .

$0 < - | x - 4 | - f (x)$

Step 4