Algebra Examples

$f (x) < - \sqrt{x + 2} - 4$

Step 1

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

$0 < - \sqrt{x + 2} - 4 - f (x)$

Step 2

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

Tap for more steps...

Step 2.1

Rewrite in slope-intercept form.

Tap for more steps...

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.

$- \sqrt{x + 2} - 4 - f (x) > 0$

Step 2.1.3

Move all terms not containing $- \sqrt{x + 2}$ to the right side of the inequality.

Tap for more steps...

Step 2.1.3.1

Add $4$ to both sides of the inequality.

$- \sqrt{x + 2} - f (x) > 4$

Step 2.1.3.2

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

$- \sqrt{x + 2} > 4 + f (x)$

Step 2.1.4

To remove the radical on the left side of the inequality, square both sides of the inequality.

${(- \sqrt{x + 2})}^{2} > {(4 + f (x))}^{2}$

Step 2.1.5

Simplify each side of the inequality.

Tap for more steps...

Step 2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 2}$ as ${(x + 2)}^{\frac{1}{2}}$ .

${(- {(x + 2)}^{\frac{1}{2}})}^{2} > {(4 + f (x))}^{2}$

Step 2.1.5.2

Simplify the left side.

Tap for more steps...

Step 2.1.5.2.1

Simplify ${(- {(x + 2)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.1.5.2.1.1

Apply the product rule to $- {(x + 2)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(x + 2)}^{\frac{1}{2}})}^{2} > {(4 + f (x))}^{2}$

Step 2.1.5.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(x + 2)}^{\frac{1}{2}})}^{2} > {(4 + f (x))}^{2}$

Step 2.1.5.2.1.3

Multiply ${({(x + 2)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(x + 2)}^{\frac{1}{2}})}^{2} > {(4 + f (x))}^{2}$

Step 2.1.5.2.1.4

Multiply the exponents in ${({(x + 2)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.1.5.2.1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x + 2)}^{\frac{1}{2} \cdot 2} > {(4 + f (x))}^{2}$

Step 2.1.5.2.1.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.5.2.1.4.2.1

Cancel the common factor.

${(x + 2)}^{\frac{1}{2} \cdot 2} > {(4 + f (x))}^{2}$

Step 2.1.5.2.1.4.2.2

Rewrite the expression.

${(x + 2)}^{1} > {(4 + f (x))}^{2}$

Step 2.1.5.2.1.5

Simplify.

$x + 2 > {(4 + f (x))}^{2}$

Step 2.1.5.3

Simplify the right side.

Tap for more steps...

Step 2.1.5.3.1

Simplify ${(4 + f (x))}^{2}$ .

Tap for more steps...

Step 2.1.5.3.1.1

Rewrite ${(4 + f (x))}^{2}$ as $(4 + f (x)) (4 + f (x))$ .

$x + 2 > (4 + f (x)) (4 + f (x))$

Step 2.1.5.3.1.2

Expand $(4 + f (x)) (4 + f (x))$ using the FOIL Method.

Tap for more steps...

Step 2.1.5.3.1.2.1

Apply the distributive property.

$x + 2 > 4 (4 + f (x)) + f (x) (4 + f (x))$

Step 2.1.5.3.1.2.2

Apply the distributive property.

$x + 2 > 4 \cdot 4 + 4 f (x) + f (x) (4 + f (x))$

Step 2.1.5.3.1.2.3

Apply the distributive property.

$x + 2 > 4 \cdot 4 + 4 f (x) + f (x) \cdot 4 + f (x) f (x)$

Step 2.1.5.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.5.3.1.3.1

Simplify each term.

Tap for more steps...

Step 2.1.5.3.1.3.1.1

Multiply $4$ by $4$ .

$x + 2 > 16 + 4 f (x) + f (x) \cdot 4 + f (x) f (x)$

Step 2.1.5.3.1.3.1.2

Move $4$ to the left of $f (x)$ .

$x + 2 > 16 + 4 f (x) + 4 \cdot f (x) + f (x) f (x)$

Step 2.1.5.3.1.3.1.3

Multiply $f (x) f (x)$ .

Tap for more steps...

Step 2.1.5.3.1.3.1.3.1

Raise $f (x)$ to the power of $1$ .

$x + 2 > 16 + 4 f (x) + 4 f (x) + {f (x)}^{1} f (x)$

Step 2.1.5.3.1.3.1.3.2

Raise $f (x)$ to the power of $1$ .

$x + 2 > 16 + 4 f (x) + 4 f (x) + {f (x)}^{1} {f (x)}^{1}$

Step 2.1.5.3.1.3.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x + 2 > 16 + 4 f (x) + 4 f (x) + {f (x)}^{1 + 1}$

Step 2.1.5.3.1.3.1.3.4

Add $1$ and $1$ .

$x + 2 > 16 + 4 f (x) + 4 f (x) + {f (x)}^{2}$

Step 2.1.5.3.1.3.2

Add $4 f (x)$ and $4 f (x)$ .

$x + 2 > 16 + 8 f (x) + {f (x)}^{2}$

Step 2.1.6

Solve for $x$ .

Tap for more steps...

Step 2.1.6.1

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

$16 + 8 f x + f x^{2} < x + 2$

Step 2.1.6.2

Subtract $x$ from both sides of the inequality.

$16 + 8 f x + f x^{2} - x < 2$

Step 2.1.6.3

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 2.1.6.3.1

Subtract $2$ from both sides of the inequality.

$16 + 8 f x + f x^{2} - x - 2 < 0$

Step 2.1.6.3.2

Subtract $2$ from $16$ .

$8 f x + f x^{2} - x + 14 < 0$

Step 2.1.6.4

Convert the inequality to an equation.

$8 f x + f x^{2} - x + 14 = 0$

Step 2.1.6.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.1.6.6

Substitute the values $a = f$ , $b = 8 f - 1$ , and $c = 14$ into the quadratic formula and solve for $x$ .

$\frac{- (8 f - 1) \pm \sqrt{{(8 f - 1)}^{2} - 4 \cdot (f \cdot 14)}}{2 f}$

Step 2.1.6.7

Simplify the numerator.

Tap for more steps...

Step 2.1.6.7.1

Apply the distributive property.

$x = \frac{- (8 f) + 1 \pm \sqrt{{(8 f - 1)}^{2} - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.2

Multiply $8$ by $- 1$ .

$x = \frac{- 8 f + 1 \pm \sqrt{{(8 f - 1)}^{2} - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.3

Multiply $- 1$ by $- 1$ .

$x = \frac{- 8 f + 1 \pm \sqrt{{(8 f - 1)}^{2} - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.4

Rewrite ${(8 f - 1)}^{2}$ as $(8 f - 1) (8 f - 1)$ .

$x = \frac{- 8 f + 1 \pm \sqrt{(8 f - 1) (8 f - 1) - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.5

Expand $(8 f - 1) (8 f - 1)$ using the FOIL Method.

Tap for more steps...

Step 2.1.6.7.5.1

Apply the distributive property.

$x = \frac{- 8 f + 1 \pm \sqrt{8 f (8 f - 1) - 1 (8 f - 1) - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.5.2

Apply the distributive property.

$x = \frac{- 8 f + 1 \pm \sqrt{8 f (8 f) + 8 f \cdot - 1 - 1 (8 f - 1) - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.5.3

Apply the distributive property.

$x = \frac{- 8 f + 1 \pm \sqrt{8 f (8 f) + 8 f \cdot - 1 - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.6

Simplify and combine like terms.

Tap for more steps...

Step 2.1.6.7.6.1

Simplify each term.

Tap for more steps...

Step 2.1.6.7.6.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{- 8 f + 1 \pm \sqrt{8 \cdot (8 f \cdot f) + 8 f \cdot - 1 - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.6.1.2

Multiply $f$ by $f$ by adding the exponents.

Tap for more steps...

Step 2.1.6.7.6.1.2.1

Move $f$ .

$x = \frac{- 8 f + 1 \pm \sqrt{8 \cdot (8 (f \cdot f)) + 8 f \cdot - 1 - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.6.1.2.2

Multiply $f$ by $f$ .

$x = \frac{- 8 f + 1 \pm \sqrt{8 \cdot (8 f^{2}) + 8 f \cdot - 1 - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.6.1.3

Multiply $8$ by $8$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} + 8 f \cdot - 1 - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.6.1.4

Multiply $- 1$ by $8$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 8 f - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.6.1.5

Multiply $8$ by $- 1$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 8 f - 8 f - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.6.1.6

Multiply $- 1$ by $- 1$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 8 f - 8 f + 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.6.2

Subtract $8 f$ from $- 8 f$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 16 f + 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.7.7

Multiply $14$ by $- 4$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 16 f + 1 - 56 f}}{2 f}$

Step 2.1.6.7.8

Subtract $56 f$ from $- 16 f$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 2.1.6.8.1

Change the $\pm$ to $+$ .

$x = \frac{- 8 f + 1 + \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.8.2

Factor $- 1$ out of $- 8 f$ .

$x = \frac{- (8 f) + 1 + \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.8.3

Rewrite $1$ as $- 1 (- 1)$ .

$x = \frac{- (8 f) - 1 \cdot - 1 + \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.8.4

Factor $- 1$ out of $- (8 f) - 1 (- 1)$ .

$x = \frac{- (8 f - 1) + \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.8.5

Factor $- 1$ out of $\sqrt{64 f^{2} - 72 f + 1}$ .

$x = \frac{- (8 f - 1) - 1 (- \sqrt{64 f^{2} - 72 f + 1})}{2 f}$

Step 2.1.6.8.6

Factor $- 1$ out of $- (8 f - 1) - 1 (- \sqrt{64 f^{2} - 72 f + 1})$ .

$x = \frac{- (8 f - 1 - \sqrt{64 f^{2} - 72 f + 1})}{2 f}$

Step 2.1.6.8.7

Rewrite $- (8 f - 1 - \sqrt{64 f^{2} - 72 f + 1})$ as $- 1 (8 f - 1 - \sqrt{64 f^{2} - 72 f + 1})$ .

$x = \frac{- 1 (8 f - 1 - \sqrt{64 f^{2} - 72 f + 1})}{2 f}$

Step 2.1.6.8.8

Move the negative in front of the fraction.

$x = - \frac{8 f - 1 - \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 2.1.6.9.1

Simplify the numerator.

Tap for more steps...

Step 2.1.6.9.1.1

Apply the distributive property.

$x = \frac{- (8 f) + 1 \pm \sqrt{{(8 f - 1)}^{2} - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.2

Multiply $8$ by $- 1$ .

$x = \frac{- 8 f + 1 \pm \sqrt{{(8 f - 1)}^{2} - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.3

Multiply $- 1$ by $- 1$ .

$x = \frac{- 8 f + 1 \pm \sqrt{{(8 f - 1)}^{2} - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.4

Rewrite ${(8 f - 1)}^{2}$ as $(8 f - 1) (8 f - 1)$ .

$x = \frac{- 8 f + 1 \pm \sqrt{(8 f - 1) (8 f - 1) - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.5

Expand $(8 f - 1) (8 f - 1)$ using the FOIL Method.

Tap for more steps...

Step 2.1.6.9.1.5.1

Apply the distributive property.

$x = \frac{- 8 f + 1 \pm \sqrt{8 f (8 f - 1) - 1 (8 f - 1) - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.5.2

Apply the distributive property.

$x = \frac{- 8 f + 1 \pm \sqrt{8 f (8 f) + 8 f \cdot - 1 - 1 (8 f - 1) - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.5.3

Apply the distributive property.

$x = \frac{- 8 f + 1 \pm \sqrt{8 f (8 f) + 8 f \cdot - 1 - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.6

Simplify and combine like terms.

Tap for more steps...

Step 2.1.6.9.1.6.1

Simplify each term.

Tap for more steps...

Step 2.1.6.9.1.6.1.1

Rewrite using the commutative property of multiplication.

$x = \frac{- 8 f + 1 \pm \sqrt{8 \cdot (8 f \cdot f) + 8 f \cdot - 1 - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.6.1.2

Multiply $f$ by $f$ by adding the exponents.

Tap for more steps...

Step 2.1.6.9.1.6.1.2.1

Move $f$ .

$x = \frac{- 8 f + 1 \pm \sqrt{8 \cdot (8 (f \cdot f)) + 8 f \cdot - 1 - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.6.1.2.2

Multiply $f$ by $f$ .

$x = \frac{- 8 f + 1 \pm \sqrt{8 \cdot (8 f^{2}) + 8 f \cdot - 1 - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.6.1.3

Multiply $8$ by $8$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} + 8 f \cdot - 1 - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.6.1.4

Multiply $- 1$ by $8$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 8 f - 1 (8 f) - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.6.1.5

Multiply $8$ by $- 1$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 8 f - 8 f - 1 \cdot - 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.6.1.6

Multiply $- 1$ by $- 1$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 8 f - 8 f + 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.6.2

Subtract $8 f$ from $- 8 f$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 16 f + 1 - 4 \cdot f \cdot 14}}{2 f}$

Step 2.1.6.9.1.7

Multiply $14$ by $- 4$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 16 f + 1 - 56 f}}{2 f}$

Step 2.1.6.9.1.8

Subtract $56 f$ from $- 16 f$ .

$x = \frac{- 8 f + 1 \pm \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.9.2

Change the $\pm$ to $-$ .

$x = \frac{- 8 f + 1 - \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.9.3

Factor $- 1$ out of $- 8 f$ .

$x = \frac{- (8 f) + 1 - \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.9.4

Rewrite $1$ as $- 1 (- 1)$ .

$x = \frac{- (8 f) - 1 \cdot - 1 - \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.9.5

Factor $- 1$ out of $- (8 f) - 1 (- 1)$ .

$x = \frac{- (8 f - 1) - \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.9.6

Factor $- 1$ out of $- \sqrt{64 f^{2} - 72 f + 1}$ .

$x = \frac{- (8 f - 1) - (\sqrt{64 f^{2} - 72 f + 1})}{2 f}$

Step 2.1.6.9.7

Factor $- 1$ out of $- (8 f - 1) - (\sqrt{64 f^{2} - 72 f + 1})$ .

$x = \frac{- (8 f - 1 + \sqrt{64 f^{2} - 72 f + 1})}{2 f}$

Step 2.1.6.9.8

Rewrite $- (8 f - 1 + \sqrt{64 f^{2} - 72 f + 1})$ as $- 1 (8 f - 1 + \sqrt{64 f^{2} - 72 f + 1})$ .

$x = \frac{- 1 (8 f - 1 + \sqrt{64 f^{2} - 72 f + 1})}{2 f}$

Step 2.1.6.9.9

Move the negative in front of the fraction.

$x = - \frac{8 f - 1 + \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.6.10

Consolidate the solutions.

$x = - \frac{8 f - 1 - \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

$x = - \frac{8 f - 1 + \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

$x = - \frac{8 f - 1 - \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

$x = - \frac{8 f - 1 + \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

Step 2.1.7

Rewrite in slope-intercept form.

$= - \frac{8 - 1 - \sqrt{64^{2} - 72 + 1}}{2}$

$= f - \frac{8 f - 1 + \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

$= - \frac{8 - 1 - \sqrt{64^{2} - 72 + 1}}{2}$

$= f - \frac{8 f - 1 + \sqrt{64 f^{2} - 72 f + 1}}{2 f}$

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 $- \sqrt{x + 2} - 4 - f (x)$ .

$0 < - \sqrt{x + 2} - 4 - f (x)$

Step 4