Pre-Algebra Examples

$3 x + \frac{\frac{1}{2}}{3 \cdot x} + 4 x - 2 + x = 169$

Step 1

Factor each term.

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Step 1.1

Multiply the numerator by the reciprocal of the denominator.

$3 x + \frac{1}{2} \cdot \frac{1}{3 \cdot x} + 4 x - 2 + x = 169$

Step 1.2

Multiply $3$ by $x$ .

$3 x + \frac{1}{2} \cdot \frac{1}{3 x} + 4 x - 2 + x = 169$

Step 1.3

Multiply $\frac{1}{2} \cdot \frac{1}{3 x}$ .

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Step 1.3.1

Multiply $\frac{1}{2}$ by $\frac{1}{3 x}$ .

$3 x + \frac{1}{2 (3 x)} + 4 x - 2 + x = 169$

Step 1.3.2

Multiply $3$ by $2$ .

$3 x + \frac{1}{6 x} + 4 x - 2 + x = 169$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 6 x, 1, 1, 1, 1$

Step 2.2

The LCM of one and any expression is the expression.

$6 x$

Step 3

Multiply each term in $3 x + \frac{1}{6 x} + 4 x - 2 + x = 169$ by $6 x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $3 x + \frac{1}{6 x} + 4 x - 2 + x = 169$ by $6 x$ .

$3 x (6 x) + \frac{1}{6 x} (6 x) + 4 x (6 x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Rewrite using the commutative property of multiplication.

$3 \cdot 6 x \cdot x + \frac{1}{6 x} (6 x) + 4 x (6 x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.1.2.1

Move $x$ .

$3 \cdot 6 (x \cdot x) + \frac{1}{6 x} (6 x) + 4 x (6 x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.2.2

Multiply $x$ by $x$ .

$3 \cdot 6 x^{2} + \frac{1}{6 x} (6 x) + 4 x (6 x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.3

Multiply $3$ by $6$ .

$18 x^{2} + \frac{1}{6 x} (6 x) + 4 x (6 x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.4

Rewrite using the commutative property of multiplication.

$18 x^{2} + 6 \frac{1}{6 x} x + 4 x (6 x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.5

Cancel the common factor of $6$ .

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Step 3.2.1.5.1

Factor $6$ out of $6 x$ .

$18 x^{2} + 6 \frac{1}{6 (x)} x + 4 x (6 x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.5.2

Cancel the common factor.

$18 x^{2} + 6 \frac{1}{6 x} x + 4 x (6 x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.5.3

Rewrite the expression.

$18 x^{2} + \frac{1}{x} x + 4 x (6 x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.6

Cancel the common factor of $x$ .

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Step 3.2.1.6.1

Cancel the common factor.

$18 x^{2} + \frac{1}{x} x + 4 x (6 x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.6.2

Rewrite the expression.

$18 x^{2} + 1 + 4 x (6 x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.7

Rewrite using the commutative property of multiplication.

$18 x^{2} + 1 + 4 \cdot 6 x \cdot x - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.8

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.1.8.1

Move $x$ .

$18 x^{2} + 1 + 4 \cdot 6 (x \cdot x) - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.8.2

Multiply $x$ by $x$ .

$18 x^{2} + 1 + 4 \cdot 6 x^{2} - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.9

Multiply $4$ by $6$ .

$18 x^{2} + 1 + 24 x^{2} - 2 (6 x) + x (6 x) = 169 (6 x)$

Step 3.2.1.10

Multiply $6$ by $- 2$ .

$18 x^{2} + 1 + 24 x^{2} - 12 x + x (6 x) = 169 (6 x)$

Step 3.2.1.11

Rewrite using the commutative property of multiplication.

$18 x^{2} + 1 + 24 x^{2} - 12 x + 6 x \cdot x = 169 (6 x)$

Step 3.2.1.12

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.1.12.1

Move $x$ .

$18 x^{2} + 1 + 24 x^{2} - 12 x + 6 (x \cdot x) = 169 (6 x)$

Step 3.2.1.12.2

Multiply $x$ by $x$ .

$18 x^{2} + 1 + 24 x^{2} - 12 x + 6 x^{2} = 169 (6 x)$

Step 3.2.2

Simplify by adding terms.

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Step 3.2.2.1

Add $18 x^{2}$ and $24 x^{2}$ .

$42 x^{2} + 1 - 12 x + 6 x^{2} = 169 (6 x)$

Step 3.2.2.2

Add $42 x^{2}$ and $6 x^{2}$ .

$48 x^{2} + 1 - 12 x = 169 (6 x)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $6$ by $169$ .

$48 x^{2} + 1 - 12 x = 1014 x$

Step 4

Solve the equation.

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $1014 x$ from both sides of the equation.

$48 x^{2} + 1 - 12 x - 1014 x = 0$

Step 4.1.2

Subtract $1014 x$ from $- 12 x$ .

$48 x^{2} + 1 - 1026 x = 0$

Step 4.2

Use the quadratic formula to find the solutions.

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

Step 4.3

Substitute the values $a = 48$ , $b = - 1026$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{1026 \pm \sqrt{{(- 1026)}^{2} - 4 \cdot (48 \cdot 1)}}{2 \cdot 48}$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Raise $- 1026$ to the power of $2$ .

$x = \frac{1026 \pm \sqrt{1052676 - 4 \cdot 48 \cdot 1}}{2 \cdot 48}$

Step 4.4.1.2

Multiply $- 4 \cdot 48 \cdot 1$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $48$ .

$x = \frac{1026 \pm \sqrt{1052676 - 192 \cdot 1}}{2 \cdot 48}$

Step 4.4.1.2.2

Multiply $- 192$ by $1$ .

$x = \frac{1026 \pm \sqrt{1052676 - 192}}{2 \cdot 48}$

Step 4.4.1.3

Subtract $192$ from $1052676$ .

$x = \frac{1026 \pm \sqrt{1052484}}{2 \cdot 48}$

Step 4.4.1.4

Rewrite $1052484$ as $2^{2} \cdot 263121$ .

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Step 4.4.1.4.1

Factor $4$ out of $1052484$ .

$x = \frac{1026 \pm \sqrt{4 (263121)}}{2 \cdot 48}$

Step 4.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{1026 \pm \sqrt{2^{2} \cdot 263121}}{2 \cdot 48}$

Step 4.4.1.5

Pull terms out from under the radical.

$x = \frac{1026 \pm 2 \sqrt{263121}}{2 \cdot 48}$

Step 4.4.2

Multiply $2$ by $48$ .

$x = \frac{1026 \pm 2 \sqrt{263121}}{96}$

Step 4.4.3

Simplify $\frac{1026 \pm 2 \sqrt{263121}}{96}$ .

$x = \frac{513 \pm \sqrt{263121}}{48}$

Step 4.5

The final answer is the combination of both solutions.

$x = \frac{513 + \sqrt{263121}}{48}, \frac{513 - \sqrt{263121}}{48}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{513 + \sqrt{263121}}{48}, \frac{513 - \sqrt{263121}}{48}$

Decimal Form:

$x = 21.37402529 \dots, 0.00097470 \dots$