Algebra Examples

$6 (\frac{x + 1}{8} - \frac{2 x - 3}{16}) = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} (3 x - 2)$

Step 1

Simplify $6 (\frac{x + 1}{8} - \frac{2 x - 3}{16})$ .

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Step 1.1

To write $\frac{x + 1}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 (\frac{x + 1}{8} \cdot \frac{2}{2} - \frac{2 x - 3}{16}) = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.2

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

Multiply $\frac{x + 1}{8}$ by $\frac{2}{2}$ .

$6 (\frac{(x + 1) \cdot 2}{8 \cdot 2} - \frac{2 x - 3}{16}) = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.2.2

Multiply $8$ by $2$ .

$6 (\frac{(x + 1) \cdot 2}{16} - \frac{2 x - 3}{16}) = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.3

Combine the numerators over the common denominator.

$6 \frac{(x + 1) \cdot 2 - (2 x - 3)}{16} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$6 \frac{x \cdot 2 + 1 \cdot 2 - (2 x - 3)}{16} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.4.2

Move $2$ to the left of $x$ .

$6 \frac{2 \cdot x + 1 \cdot 2 - (2 x - 3)}{16} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.4.3

Multiply $2$ by $1$ .

$6 \frac{2 \cdot x + 2 - (2 x - 3)}{16} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.4.4

Apply the distributive property.

$6 \frac{2 x + 2 - (2 x) - - 3}{16} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.4.5

Multiply $2$ by $- 1$ .

$6 \frac{2 x + 2 - 2 x - - 3}{16} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.4.6

Multiply $- 1$ by $- 3$ .

$6 \frac{2 x + 2 - 2 x + 3}{16} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.4.7

Subtract $2 x$ from $2 x$ .

$6 \frac{0 + 2 + 3}{16} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.4.8

Add $0$ and $2$ .

$6 \frac{2 + 3}{16} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.4.9

Add $2$ and $3$ .

$6 (\frac{5}{16}) = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.5

Simplify terms.

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Step 1.5.1

Cancel the common factor of $2$ .

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Step 1.5.1.1

Factor $2$ out of $6$ .

$2 (3) \frac{5}{16} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.5.1.2

Factor $2$ out of $16$ .

$2 \cdot 3 \frac{5}{2 \cdot 8} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.5.1.3

Cancel the common factor.

$2 \cdot 3 \frac{5}{2 \cdot 8} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.5.1.4

Rewrite the expression.

$3 (\frac{5}{8}) = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.5.2

Combine $3$ and $\frac{5}{8}$ .

$\frac{3 \cdot 5}{8} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 1.5.3

Multiply $3$ by $5$ .

$\frac{15}{8} = 3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 2

Simplify $3 (\frac{3}{4} x - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $\frac{3}{4}$ and $x$ .

$\frac{15}{8} = 3 (\frac{3 x}{4} - \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 2.1.2

Apply the distributive property.

$\frac{15}{8} = 3 \frac{3 x}{4} + 3 (- \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 2.1.3

Multiply $3 \frac{3 x}{4}$ .

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Step 2.1.3.1

Combine $3$ and $\frac{3 x}{4}$ .

$\frac{15}{8} = \frac{3 (3 x)}{4} + 3 (- \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 2.1.3.2

Multiply $3$ by $3$ .

$\frac{15}{8} = \frac{9 x}{4} + 3 (- \frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 2.1.4

Multiply $3 (- \frac{1}{4})$ .

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Step 2.1.4.1

Multiply $- 1$ by $3$ .

$\frac{15}{8} = \frac{9 x}{4} - 3 (\frac{1}{4}) - \frac{3}{8} \cdot (3 x - 2)$

Step 2.1.4.2

Combine $- 3$ and $\frac{1}{4}$ .

$\frac{15}{8} = \frac{9 x}{4} + \frac{- 3}{4} - \frac{3}{8} \cdot (3 x - 2)$

Step 2.1.5

Move the negative in front of the fraction.

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} - \frac{3}{8} \cdot (3 x - 2)$

Step 2.1.6

Apply the distributive property.

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} - \frac{3}{8} (3 x) - \frac{3}{8} \cdot - 2$

Step 2.1.7

Multiply $- \frac{3}{8} (3 x)$ .

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Step 2.1.7.1

Multiply $3$ by $- 1$ .

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} - 3 (\frac{3}{8}) x - \frac{3}{8} \cdot - 2$

Step 2.1.7.2

Combine $- 3$ and $\frac{3}{8}$ .

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} + \frac{- 3 \cdot 3}{8} x - \frac{3}{8} \cdot - 2$

Step 2.1.7.3

Multiply $- 3$ by $3$ .

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} + \frac{- 9}{8} x - \frac{3}{8} \cdot - 2$

Step 2.1.7.4

Combine $\frac{- 9}{8}$ and $x$ .

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} + \frac{- 9 x}{8} - \frac{3}{8} \cdot - 2$

Step 2.1.8

Cancel the common factor of $2$ .

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Step 2.1.8.1

Move the leading negative in $- \frac{3}{8}$ into the numerator.

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} + \frac{- 9 x}{8} + \frac{- 3}{8} \cdot - 2$

Step 2.1.8.2

Factor $2$ out of $8$ .

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} + \frac{- 9 x}{8} + \frac{- 3}{2 (4)} \cdot - 2$

Step 2.1.8.3

Factor $2$ out of $- 2$ .

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} + \frac{- 9 x}{8} + \frac{- 3}{2 \cdot 4} \cdot (2 \cdot - 1)$

Step 2.1.8.4

Cancel the common factor.

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} + \frac{- 9 x}{8} + \frac{- 3}{2 \cdot 4} \cdot (2 \cdot - 1)$

Step 2.1.8.5

Rewrite the expression.

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} + \frac{- 9 x}{8} + \frac{- 3}{4} \cdot - 1$

Step 2.1.9

Combine $\frac{- 3}{4}$ and $- 1$ .

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} + \frac{- 9 x}{8} + \frac{- 3 \cdot - 1}{4}$

Step 2.1.10

Multiply $- 3$ by $- 1$ .

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} + \frac{- 9 x}{8} + \frac{3}{4}$

Step 2.1.11

Move the negative in front of the fraction.

$\frac{15}{8} = \frac{9 x}{4} - \frac{3}{4} - \frac{9 x}{8} + \frac{3}{4}$

Step 2.2

Combine the opposite terms in $\frac{9 x}{4} - \frac{3}{4} - \frac{9 x}{8} + \frac{3}{4}$ .

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Step 2.2.1

Add $- \frac{3}{4}$ and $\frac{3}{4}$ .

$\frac{15}{8} = \frac{9 x}{4} - \frac{9 x}{8} + 0$

Step 2.2.2

Add $\frac{9 x}{4} - \frac{9 x}{8}$ and $0$ .

$\frac{15}{8} = \frac{9 x}{4} - \frac{9 x}{8}$

Step 2.3

To write $\frac{9 x}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{15}{8} = \frac{9 x}{4} \cdot \frac{2}{2} - \frac{9 x}{8}$

Step 2.4

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{9 x}{4}$ by $\frac{2}{2}$ .

$\frac{15}{8} = \frac{9 x \cdot 2}{4 \cdot 2} - \frac{9 x}{8}$

Step 2.4.2

Multiply $4$ by $2$ .

$\frac{15}{8} = \frac{9 x \cdot 2}{8} - \frac{9 x}{8}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{15}{8} = \frac{9 x \cdot 2 - 9 x}{8}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Factor $9 x$ out of $9 x \cdot 2 - 9 x$ .

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Step 2.6.1.1

Factor $9 x$ out of $9 x \cdot 2$ .

$\frac{15}{8} = \frac{9 x (2) - 9 x}{8}$

Step 2.6.1.2

Factor $9 x$ out of $- 9 x$ .

$\frac{15}{8} = \frac{9 x (2) + 9 x (- 1)}{8}$

Step 2.6.1.3

Factor $9 x$ out of $9 x (2) + 9 x (- 1)$ .

$\frac{15}{8} = \frac{9 x (2 - 1)}{8}$

Step 2.6.2

Subtract $1$ from $2$ .

$\frac{15}{8} = \frac{9 x \cdot 1}{8}$

Step 2.6.3

Multiply $9$ by $1$ .

$\frac{15}{8} = \frac{9 x}{8}$

Step 3

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{9 x}{8} = \frac{15}{8}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$9 x = 15$

Step 5

Divide each term in $9 x = 15$ by $9$ and simplify.

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Step 5.1

Divide each term in $9 x = 15$ by $9$ .

$\frac{9 x}{9} = \frac{15}{9}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $9$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{15}{9}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{15}{9}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $15$ and $9$ .

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Step 5.3.1.1

Factor $3$ out of $15$ .

$x = \frac{3 (5)}{9}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $3$ out of $9$ .

$x = \frac{3 \cdot 5}{3 \cdot 3}$

Step 5.3.1.2.2

Cancel the common factor.

$x = \frac{3 \cdot 5}{3 \cdot 3}$

Step 5.3.1.2.3

Rewrite the expression.

$x = \frac{5}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{3}$

Decimal Form:

$x = 1.\overline{6}$

Mixed Number Form:

$x = 1 \frac{2}{3}$