Algebra Examples

$1 \frac{2}{3} \div 5.5 + 1 \frac{7}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1

Simplify the left side.

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Step 1.1

Simplify $1 \frac{2}{3} \div 5.5 + 1 \frac{7}{12} \div (2.25 - x)$ .

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Step 1.1.1

Convert $1 \frac{2}{3}$ to an improper fraction.

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Step 1.1.1.1

A mixed number is an addition of its whole and fractional parts.

$(1 + \frac{2}{3}) \div 5.5 + 1 \frac{7}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.1.2

Add $1$ and $\frac{2}{3}$ .

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Step 1.1.1.2.1

Write $1$ as a fraction with a common denominator.

$(\frac{3}{3} + \frac{2}{3}) \div 5.5 + 1 \frac{7}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.1.2.2

Combine the numerators over the common denominator.

$\frac{3 + 2}{3} \div 5.5 + 1 \frac{7}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.1.2.3

Add $3$ and $2$ .

$\frac{5}{3} \div 5.5 + 1 \frac{7}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.2

Convert $1 \frac{7}{12}$ to an improper fraction.

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Step 1.1.2.1

A mixed number is an addition of its whole and fractional parts.

$\frac{5}{3} \div 5.5 + (1 + \frac{7}{12}) \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.2.2

Add $1$ and $\frac{7}{12}$ .

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Step 1.1.2.2.1

Write $1$ as a fraction with a common denominator.

$\frac{5}{3} \div 5.5 + (\frac{12}{12} + \frac{7}{12}) \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.2.2.2

Combine the numerators over the common denominator.

$\frac{5}{3} \div 5.5 + \frac{12 + 7}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.2.2.3

Add $12$ and $7$ .

$\frac{5}{3} \div 5.5 + \frac{19}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.3

Simplify each term.

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Step 1.1.3.1

Rewrite the division as a fraction.

$\frac{\frac{5}{3}}{5.5} + \frac{19}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.3.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{5}{3} \cdot \frac{1}{5.5} + \frac{19}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.3.3

Divide $1$ by $5.5$ .

$\frac{5}{3} \cdot 0.\overline{18} + \frac{19}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.3.4

Multiply $\frac{5}{3} \cdot 0.\overline{18}$ .

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Step 1.1.3.4.1

Combine $\frac{5}{3}$ and $0.\overline{18}$ .

$\frac{5 \cdot 0.\overline{18}}{3} + \frac{19}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.3.4.2

Multiply $5$ by $0.\overline{18}$ .

$\frac{0.\overline{90}}{3} + \frac{19}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.3.5

Divide $0.\overline{90}$ by $3$ .

$0.\overline{30} + \frac{19}{12} \div (2.25 - x) = 1 \frac{1}{6}$

Step 1.1.3.6

Rewrite the division as a fraction.

$0.\overline{30} + \frac{\frac{19}{12}}{2.25 - x} = 1 \frac{1}{6}$

Step 1.1.3.7

Multiply the numerator by the reciprocal of the denominator.

$0.\overline{30} + \frac{19}{12} \cdot \frac{1}{2.25 - x} = 1 \frac{1}{6}$

Step 1.1.3.8

Factor $0.25$ out of $2.25 - x$ .

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Step 1.1.3.8.1

Factor $0.25$ out of $2.25$ .

$0.\overline{30} + \frac{19}{12} \cdot \frac{1}{0.25 (9) - x} = 1 \frac{1}{6}$

Step 1.1.3.8.2

Factor $0.25$ out of $- x$ .

$0.\overline{30} + \frac{19}{12} \cdot \frac{1}{0.25 (9) + 0.25 (- 4 x)} = 1 \frac{1}{6}$

Step 1.1.3.8.3

Factor $0.25$ out of $0.25 (9) + 0.25 (- 4 x)$ .

$0.\overline{30} + \frac{19}{12} \cdot \frac{1}{0.25 (9 - 4 x)} = 1 \frac{1}{6}$

Step 1.1.3.9

Cancel the common factor of $0.25$ .

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Step 1.1.3.9.1

Factor $0.25$ out of $19$ .

$0.\overline{30} + \frac{0.25 (76)}{12} \cdot \frac{1}{0.25 (9 - 4 x)} = 1 \frac{1}{6}$

Step 1.1.3.9.2

Cancel the common factor.

$0.\overline{30} + \frac{0.25 \cdot 76}{12} \cdot \frac{1}{0.25 (9 - 4 x)} = 1 \frac{1}{6}$

Step 1.1.3.9.3

Rewrite the expression.

$0.\overline{30} + \frac{76}{12} \cdot \frac{1}{9 - 4 x} = 1 \frac{1}{6}$

Step 1.1.3.10

Multiply $\frac{76}{12}$ by $\frac{1}{9 - 4 x}$ .

$0.\overline{30} + \frac{76}{12 (9 - 4 x)} = 1 \frac{1}{6}$

Step 1.1.3.11

Cancel the common factor of $76$ and $12$ .

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Step 1.1.3.11.1

Factor $4$ out of $76$ .

$0.\overline{30} + \frac{4 (19)}{12 (9 - 4 x)} = 1 \frac{1}{6}$

Step 1.1.3.11.2

Cancel the common factors.

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Step 1.1.3.11.2.1

Factor $4$ out of $12 (9 - 4 x)$ .

$0.\overline{30} + \frac{4 (19)}{4 (3 (9 - 4 x))} = 1 \frac{1}{6}$

Step 1.1.3.11.2.2

Cancel the common factor.

$0.\overline{30} + \frac{4 \cdot 19}{4 (3 (9 - 4 x))} = 1 \frac{1}{6}$

Step 1.1.3.11.2.3

Rewrite the expression.

$0.\overline{30} + \frac{19}{3 (9 - 4 x)} = 1 \frac{1}{6}$

Step 2

Simplify the right side.

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Step 2.1

Convert $1 \frac{1}{6}$ to an improper fraction.

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Step 2.1.1

A mixed number is an addition of its whole and fractional parts.

$0.\overline{30} + \frac{19}{3 (9 - 4 x)} = 1 + \frac{1}{6}$

Step 2.1.2

Add $1$ and $\frac{1}{6}$ .

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Step 2.1.2.1

Write $1$ as a fraction with a common denominator.

$0.\overline{30} + \frac{19}{3 (9 - 4 x)} = \frac{6}{6} + \frac{1}{6}$

Step 2.1.2.2

Combine the numerators over the common denominator.

$0.\overline{30} + \frac{19}{3 (9 - 4 x)} = \frac{6 + 1}{6}$

Step 2.1.2.3

Add $6$ and $1$ .

$0.\overline{30} + \frac{19}{3 (9 - 4 x)} = \frac{7}{6}$

Step 3

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

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Step 3.1

Subtract $0.\overline{30}$ from both sides of the equation.

$\frac{19}{3 (9 - 4 x)} = \frac{7}{6} - 0.\overline{30}$

Step 3.2

To write $- 0.\overline{30}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{19}{3 (9 - 4 x)} = \frac{7}{6} - 0.\overline{30} \cdot \frac{6}{6}$

Step 3.3

Combine $- 0.\overline{30}$ and $\frac{6}{6}$ .

$\frac{19}{3 (9 - 4 x)} = \frac{7}{6} + \frac{- 0.\overline{30} \cdot 6}{6}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{19}{3 (9 - 4 x)} = \frac{7 - 0.\overline{30} \cdot 6}{6}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 0.\overline{30}$ by $6$ .

$\frac{19}{3 (9 - 4 x)} = \frac{7 - 1.\overline{81}}{6}$

Step 3.5.2

Subtract $1.\overline{81}$ from $7$ .

$\frac{19}{3 (9 - 4 x)} = \frac{5.\overline{18}}{6}$

Step 3.6

Divide $5.\overline{18}$ by $6$ .

$\frac{19}{3 (9 - 4 x)} = 0.8\overline{63}$

Step 4

Find the LCD of the terms in the equation.

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Step 4.1

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

$3 (9 - 4 x), 1$

Step 4.2

The LCM of one and any expression is the expression.

$3 (9 - 4 x)$

Step 5

Multiply each term in $\frac{19}{3 (9 - 4 x)} = 0.8\overline{63}$ by $3 (9 - 4 x)$ to eliminate the fractions.

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Step 5.1

Multiply each term in $\frac{19}{3 (9 - 4 x)} = 0.8\overline{63}$ by $3 (9 - 4 x)$ .

$\frac{19}{3 (9 - 4 x)} (3 (9 - 4 x)) = 0.8\overline{63} (3 (9 - 4 x))$

Step 5.2

Simplify the left side.

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Step 5.2.1

Rewrite using the commutative property of multiplication.

$3 \frac{19}{3 (9 - 4 x)} (9 - 4 x) = 0.8\overline{63} (3 (9 - 4 x))$

Step 5.2.2

Cancel the common factor of $3$ .

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Step 5.2.2.1

Cancel the common factor.

$3 \frac{19}{3 (9 - 4 x)} (9 - 4 x) = 0.8\overline{63} (3 (9 - 4 x))$

Step 5.2.2.2

Rewrite the expression.

$\frac{19}{9 - 4 x} (9 - 4 x) = 0.8\overline{63} (3 (9 - 4 x))$

Step 5.2.3

Cancel the common factor of $9 - 4 x$ .

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Step 5.2.3.1

Cancel the common factor.

$\frac{19}{9 - 4 x} (9 - 4 x) = 0.8\overline{63} (3 (9 - 4 x))$

Step 5.2.3.2

Rewrite the expression.

$19 = 0.8\overline{63} (3 (9 - 4 x))$

Step 5.3

Simplify the right side.

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Step 5.3.1

Apply the distributive property.

$19 = 0.8\overline{63} (3 \cdot 9 + 3 (- 4 x))$

Step 5.3.2

Multiply.

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Step 5.3.2.1

Multiply $3$ by $9$ .

$19 = 0.8\overline{63} (27 + 3 (- 4 x))$

Step 5.3.2.2

Multiply $- 4$ by $3$ .

$19 = 0.8\overline{63} (27 - 12 x)$

Step 5.3.3

Apply the distributive property.

$19 = 0.8\overline{63} \cdot 27 + 0.8\overline{63} (- 12 x)$

Step 5.3.4

Multiply.

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Step 5.3.4.1

Multiply $0.8\overline{63}$ by $27$ .

$19 = 23.3\overline{18} + 0.8\overline{63} (- 12 x)$

Step 5.3.4.2

Multiply $- 12$ by $0.8\overline{63}$ .

$19 = 23.3\overline{18} - 10.\overline{36} x$

Step 6

Solve the equation.

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Step 6.1

Rewrite the equation as $23.3\overline{18} - 10.\overline{36} x = 19$ .

$23.3\overline{18} - 10.\overline{36} x = 19$

Step 6.2

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

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Step 6.2.1

Subtract $23.3\overline{18}$ from both sides of the equation.

$- 10.\overline{36} x = 19 - 23.3\overline{18}$

Step 6.2.2

Subtract $23.3\overline{18}$ from $19$ .

$- 10.\overline{36} x = - 4.3\overline{18}$

Step 6.3

Divide each term in $- 10.\overline{36} x = - 4.3\overline{18}$ by $- 10.\overline{36}$ and simplify.

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Step 6.3.1

Divide each term in $- 10.\overline{36} x = - 4.3\overline{18}$ by $- 10.\overline{36}$ .

$\frac{- 10.\overline{36} x}{- 10.\overline{36}} = \frac{- 4.3\overline{18}}{- 10.\overline{36}}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $- 10.\overline{36}$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{- 10.\overline{36} x}{- 10.\overline{36}} = \frac{- 4.3\overline{18}}{- 10.\overline{36}}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4.3\overline{18}}{- 10.\overline{36}}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $- 4.3\overline{18}$ by $- 10.\overline{36}$ .

$x = 0.41\overline{6}$