Pre-Algebra Examples

$\frac{8}{\sqrt{56 - 7 x}}$

Step 1

Find the domain for $y = \frac{8}{\sqrt{56 - 7 x}}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 1.1

Set the radicand in $\sqrt{7 (8 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$7 (8 - x) \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Divide each term in $7 (8 - x) \geq 0$ by $7$ and simplify.

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Step 1.2.1.1

Divide each term in $7 (8 - x) \geq 0$ by $7$ .

$\frac{7 (8 - x)}{7} \geq \frac{0}{7}$

Step 1.2.1.2

Simplify the left side.

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Step 1.2.1.2.1

Cancel the common factor of $7$ .

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Step 1.2.1.2.1.1

Cancel the common factor.

$\frac{7 (8 - x)}{7} \geq \frac{0}{7}$

Step 1.2.1.2.1.2

Divide $8 - x$ by $1$ .

$8 - x \geq \frac{0}{7}$

Step 1.2.1.3

Simplify the right side.

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Step 1.2.1.3.1

Divide $0$ by $7$ .

$8 - x \geq 0$

Step 1.2.2

Subtract $8$ from both sides of the inequality.

$- x \geq - 8$

Step 1.2.3

Divide each term in $- x \geq - 8$ by $- 1$ and simplify.

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Step 1.2.3.1

Divide each term in $- x \geq - 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 8}{- 1}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 8}{- 1}$

Step 1.2.3.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 8}{- 1}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $- 8$ by $- 1$ .

$x \leq 8$

Step 1.3

Set the denominator in $\frac{8 \sqrt{7 (8 - x)}}{7 (8 - x)}$ equal to $0$ to find where the expression is undefined.

$7 (8 - x) = 0$

Step 1.4

Solve for $x$ .

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Step 1.4.1

Divide each term in $7 (8 - x) = 0$ by $7$ and simplify.

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Step 1.4.1.1

Divide each term in $7 (8 - x) = 0$ by $7$ .

$\frac{7 (8 - x)}{7} = \frac{0}{7}$

Step 1.4.1.2

Simplify the left side.

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Step 1.4.1.2.1

Cancel the common factor of $7$ .

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Step 1.4.1.2.1.1

Cancel the common factor.

$\frac{7 (8 - x)}{7} = \frac{0}{7}$

Step 1.4.1.2.1.2

Divide $8 - x$ by $1$ .

$8 - x = \frac{0}{7}$

Step 1.4.1.3

Simplify the right side.

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Step 1.4.1.3.1

Divide $0$ by $7$ .

$8 - x = 0$

Step 1.4.2

Subtract $8$ from both sides of the equation.

$- x = - 8$

Step 1.4.3

Divide each term in $- x = - 8$ by $- 1$ and simplify.

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Step 1.4.3.1

Divide each term in $- x = - 8$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 8}{- 1}$

Step 1.4.3.2

Simplify the left side.

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Step 1.4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 8}{- 1}$

Step 1.4.3.2.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 1}$

Step 1.4.3.3

Simplify the right side.

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Step 1.4.3.3.1

Divide $- 8$ by $- 1$ .

$x = 8$

Step 1.5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 8)$

Set-Builder Notation:

${x | x < 8}$

Interval Notation:

$(- \infty, 8)$

Set-Builder Notation:

${x | x < 8}$

Step 2

To find the radical expression end point, substitute the $x$ value $8$ , which is the least value in the domain, into $f (x) = \frac{8 \sqrt{7 (8 - x)}}{7 (8 - x)}$ .

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Step 2.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = \frac{8 \sqrt{7 (8 - (8))}}{7 (8 - (8))}$

Step 2.2

Multiply $- 1$ by $8$ .

$f (8) = \frac{8 \sqrt{7 (8 - (8))}}{7 (8 - 8)}$

Step 2.3

Subtract $8$ from $8$ .

$f (8) = \frac{8 \sqrt{7 (8 - (8))}}{7 \cdot 0}$

Step 2.4

Multiply $7$ by $0$ .

$f (8) = \frac{8 \sqrt{7 (8 - (8))}}{0}$

Step 2.5

The expression contains a division by $0$ . The expression is undefined.

$f (8) = Undefined$

Undefined

Step 3

The radical expression end point is $(8, Undefined)$ .

$(8, Undefined)$

Step 4

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

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Step 4.1

Substitute the $x$ value $6$ into $f (x) = \frac{8 \sqrt{7 (8 - x)}}{7 (8 - x)}$ . In this case, the point is $(6, \frac{4 \sqrt{14}}{7})$ .

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Step 4.1.1

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{8 \sqrt{7 (8 - (6))}}{7 (8 - (6))}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Cancel the common factor of $8$ and $8 - (6)$ .

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Step 4.1.2.1.1

Rewrite $8$ as $- 1 (- 8)$ .

$f (6) = \frac{8 \sqrt{7 (8 - (6))}}{7 (- 1 \cdot - 8 - (6))}$

Step 4.1.2.1.2

Factor $- 1$ out of $- 1 (- 8) - (6)$ .

$f (6) = \frac{8 \sqrt{7 (8 - (6))}}{7 (- 1 (- 8 + 6))}$

Step 4.1.2.1.3

Factor $2$ out of $8 \sqrt{7 (8 - (6))}$ .

$f (6) = \frac{2 (4 \sqrt{7 (8 - (6))})}{7 (- 1 (- 8 + 6))}$

Step 4.1.2.1.4

Cancel the common factors.

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Step 4.1.2.1.4.1

Factor $2$ out of $7 (- 1 (- 8 + 6))$ .

$f (6) = \frac{2 (4 \sqrt{7 (8 - (6))})}{2 (7 (- 1 (- 4 + 3)))}$

Step 4.1.2.1.4.2

Cancel the common factor.

$f (6) = \frac{2 (4 \sqrt{7 (8 - (6))})}{2 (7 (- 1 (- 4 + 3)))}$

Step 4.1.2.1.4.3

Rewrite the expression.

$f (6) = \frac{4 \sqrt{7 (8 - (6))}}{7 (- 1 (- 4 + 3))}$

Step 4.1.2.2

Simplify the numerator.

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Step 4.1.2.2.1

Multiply $- 1$ by $6$ .

$f (6) = \frac{4 \sqrt{7 (8 - 6)}}{7 (- 1 (- 4 + 3))}$

Step 4.1.2.2.2

Subtract $6$ from $8$ .

$f (6) = \frac{4 \sqrt{7 \cdot 2}}{7 (- 1 (- 4 + 3))}$

Step 4.1.2.2.3

Multiply $7$ by $2$ .

$f (6) = \frac{4 \sqrt{14}}{7 (- 1 (- 4 + 3))}$

Step 4.1.2.3

Simplify the expression.

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Step 4.1.2.3.1

Multiply $- 1$ by $7$ .

$f (6) = \frac{4 \sqrt{14}}{- 7 (- 4 + 3)}$

Step 4.1.2.3.2

Add $- 4$ and $3$ .

$f (6) = \frac{4 \sqrt{14}}{- 7 \cdot - 1}$

Step 4.1.2.3.3

Multiply $- 7$ by $- 1$ .

$f (6) = \frac{4 \sqrt{14}}{7}$

Step 4.1.2.4

The final answer is $\frac{4 \sqrt{14}}{7}$ .

$y = \frac{4 \sqrt{14}}{7}$

Step 4.2

Substitute the $x$ value $7$ into $f (x) = \frac{8 \sqrt{7 (8 - x)}}{7 (8 - x)}$ . In this case, the point is $(7, \frac{8 \sqrt{7}}{7})$ .

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Step 4.2.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = \frac{8 \sqrt{7 (8 - (7))}}{7 (8 - (7))}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Multiply $- 1$ by $7$ .

$f (7) = \frac{8 \sqrt{7 (8 - 7)}}{7 (8 - (7))}$

Step 4.2.2.1.2

Subtract $7$ from $8$ .

$f (7) = \frac{8 \sqrt{7 \cdot 1}}{7 (8 - (7))}$

Step 4.2.2.1.3

Multiply $7$ by $1$ .

$f (7) = \frac{8 \sqrt{7}}{7 (8 - (7))}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

Multiply $- 1$ by $7$ .

$f (7) = \frac{8 \sqrt{7}}{7 (8 - 7)}$

Step 4.2.2.2.2

Subtract $7$ from $8$ .

$f (7) = \frac{8 \sqrt{7}}{7 \cdot 1}$

Step 4.2.2.3

Multiply $7$ by $1$ .

$f (7) = \frac{8 \sqrt{7}}{7}$

Step 4.2.2.4

The final answer is $\frac{8 \sqrt{7}}{7}$ .

$y = \frac{8 \sqrt{7}}{7}$

Step 4.3

The square root can be graphed using the points around the vertex $(8, Undefined), (6, 2.14), (7, 3.02)$

$\begin{array}{c} x & y \\ 6 & 2.14 \\ 7 & 3.02 \\ 8 & Undefined \end{array}$

Step 5