Algebra Examples

$f (x) < - 2 | x | + 6$

Step 1

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

$0 < - 2 | x | + 6 - f (x)$

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.

$- 2 | x | + 6 - f (x) > 0$

Step 2.1.3

Write $- 2 | x | + 6 - f (x) > 0$ 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.

$x \geq 0$

Step 2.1.3.2

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

$- 2 x + 6 - f (x) > 0$

Step 2.1.3.3

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

$x < 0$

Step 2.1.3.4

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

$- 2 (- x) + 6 - f (x) > 0$

Step 2.1.3.5

Write as a piecewise.

${\begin{cases} - 2 x + 6 - f (x) > 0 & x \geq 0 \\ - 2 (- x) + 6 - f (x) > 0 & x < 0 \end{cases}$

Step 2.1.3.6

Multiply $- 1$ by $- 2$ .

${\begin{cases} - 2 x + 6 - f (x) > 0 & x \geq 0 \\ 2 x + 6 - f (x) > 0 & x < 0 \end{cases}$

Step 2.1.4

Solve $- 2 x + 6 - f (x) > 0$ when $x \geq 0$ .

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Step 2.1.4.1

Solve $- 2 x + 6 - f (x) > 0$ 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 $6$ from both sides of the inequality.

$- 2 x - f (x) > - 6$

Step 2.1.4.1.1.2

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

$- 2 x > - 6 + f (x)$

Step 2.1.4.1.2

Divide each term in $- 2 x > - 6 + f (x)$ by $- 2$ and simplify.

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Step 2.1.4.1.2.1

Divide each term in $- 2 x > - 6 + f (x)$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} < \frac{- 6}{- 2} + \frac{f (x)}{- 2}$

Step 2.1.4.1.2.2

Simplify the left side.

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Step 2.1.4.1.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.1.4.1.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} < \frac{- 6}{- 2} + \frac{f (x)}{- 2}$

Step 2.1.4.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 6}{- 2} + \frac{f (x)}{- 2}$

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 $- 6$ by $- 2$ .

$x < 3 + \frac{f (x)}{- 2}$

Step 2.1.4.1.2.3.1.2

Move the negative in front of the fraction.

$x < 3 - \frac{f (x)}{2}$

Step 2.1.4.2

Find the intersection of $x < 3 - \frac{f (x)}{2}$ and $x \geq 0$ .

$0 \leq x < 3 - \frac{f (x)}{2}$

Step 2.1.5

Solve $2 x + 6 - f (x) > 0$ when $x < 0$ .

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Step 2.1.5.1

Solve $2 x + 6 - f (x) > 0$ for $x$ .

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Step 2.1.5.1.1

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

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Step 2.1.5.1.1.1

Subtract $6$ from both sides of the inequality.

$2 x - f (x) > - 6$

Step 2.1.5.1.1.2

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

$2 x > - 6 + f (x)$

Step 2.1.5.1.2

Divide each term in $2 x > - 6 + f (x)$ by $2$ and simplify.

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Step 2.1.5.1.2.1

Divide each term in $2 x > - 6 + f (x)$ by $2$ .

$\frac{2 x}{2} > \frac{- 6}{2} + \frac{f (x)}{2}$

Step 2.1.5.1.2.2

Simplify the left side.

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Step 2.1.5.1.2.2.1

Cancel the common factor of $2$ .

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Step 2.1.5.1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} > \frac{- 6}{2} + \frac{f (x)}{2}$

Step 2.1.5.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 6}{2} + \frac{f (x)}{2}$

Step 2.1.5.1.2.3

Simplify the right side.

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Step 2.1.5.1.2.3.1

Divide $- 6$ by $2$ .

$x > - 3 + \frac{f (x)}{2}$

Step 2.1.5.2

Find the intersection of $x > - 3 + \frac{f (x)}{2}$ and $x < 0$ .

$- 3 + \frac{f (x)}{2} < x < 0$

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 $- 2 | x | + 6 - f (x)$ .

$0 < - 2 | x | + 6 - f (x)$

Step 4