Basic Math Examples

$| \frac{1}{2} y - 4 | = | \frac{2}{3} y - 1 |$

Step 1

Rewrite the equation as $| \frac{2}{3} y - 1 | = | \frac{1}{2} y - 4 |$ .

$| \frac{2}{3} y - 1 | = | \frac{1}{2} y - 4 |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$\frac{2}{3} y - 1 = \frac{1}{2} y - 4$

$\frac{2}{3} y - 1 = - (\frac{1}{2} y - 4)$

$- (\frac{2}{3} y - 1) = \frac{1}{2} y - 4$

$- (\frac{2}{3} y - 1) = - (\frac{1}{2} y - 4)$

Step 3

After simplifying, there are only two unique equations to be solved.

$\frac{2}{3} y - 1 = \frac{1}{2} y - 4$

$\frac{2}{3} y - 1 = - (\frac{1}{2} y - 4)$

Step 4

Solve $\frac{2}{3} y - 1 = \frac{1}{2} y - 4$ for $y$ .

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Step 4.1

Combine $\frac{2}{3}$ and $y$ .

$\frac{2 y}{3} - 1 = \frac{1}{2} y - 4$

Step 4.2

Combine $\frac{1}{2}$ and $y$ .

$\frac{2 y}{3} - 1 = \frac{y}{2} - 4$

Step 4.3

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

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Step 4.3.1

Subtract $\frac{y}{2}$ from both sides of the equation.

$\frac{2 y}{3} - 1 - \frac{y}{2} = - 4$

Step 4.3.2

To write $\frac{2 y}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 y}{3} \cdot \frac{2}{2} - \frac{y}{2} - 1 = - 4$

Step 4.3.3

To write $- \frac{y}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 y}{3} \cdot \frac{2}{2} - \frac{y}{2} \cdot \frac{3}{3} - 1 = - 4$

Step 4.3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.4.1

Multiply $\frac{2 y}{3}$ by $\frac{2}{2}$ .

$\frac{2 y \cdot 2}{3 \cdot 2} - \frac{y}{2} \cdot \frac{3}{3} - 1 = - 4$

Step 4.3.4.2

Multiply $3$ by $2$ .

$\frac{2 y \cdot 2}{6} - \frac{y}{2} \cdot \frac{3}{3} - 1 = - 4$

Step 4.3.4.3

Multiply $\frac{y}{2}$ by $\frac{3}{3}$ .

$\frac{2 y \cdot 2}{6} - \frac{y \cdot 3}{2 \cdot 3} - 1 = - 4$

Step 4.3.4.4

Multiply $2$ by $3$ .

$\frac{2 y \cdot 2}{6} - \frac{y \cdot 3}{6} - 1 = - 4$

Step 4.3.5

Combine the numerators over the common denominator.

$\frac{2 y \cdot 2 - y \cdot 3}{6} - 1 = - 4$

Step 4.3.6

Simplify each term.

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Step 4.3.6.1

Simplify the numerator.

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Step 4.3.6.1.1

Factor $y$ out of $2 y \cdot 2 - y \cdot 3$ .

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Step 4.3.6.1.1.1

Factor $y$ out of $2 y \cdot 2$ .

$\frac{y (2 \cdot 2) - y \cdot 3}{6} - 1 = - 4$

Step 4.3.6.1.1.2

Factor $y$ out of $- y \cdot 3$ .

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

Step 4.3.6.1.1.3

Factor $y$ out of $y (2 \cdot 2) + y (- 1 \cdot 3)$ .

$\frac{y (2 \cdot 2 - 1 \cdot 3)}{6} - 1 = - 4$

Step 4.3.6.1.2

Multiply $2$ by $2$ .

$\frac{y (4 - 1 \cdot 3)}{6} - 1 = - 4$

Step 4.3.6.1.3

Multiply $- 1$ by $3$ .

$\frac{y (4 - 3)}{6} - 1 = - 4$

Step 4.3.6.1.4

Subtract $3$ from $4$ .

$\frac{y \cdot 1}{6} - 1 = - 4$

Step 4.3.6.2

Multiply $y$ by $1$ .

$\frac{y}{6} - 1 = - 4$

Step 4.4

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

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Step 4.4.1

Add $1$ to both sides of the equation.

$\frac{y}{6} = - 4 + 1$

Step 4.4.2

Add $- 4$ and $1$ .

$\frac{y}{6} = - 3$

Step 4.5

Multiply both sides of the equation by $6$ .

$6 \frac{y}{6} = 6 \cdot - 3$

Step 4.6

Simplify both sides of the equation.

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Step 4.6.1

Simplify the left side.

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Step 4.6.1.1

Cancel the common factor of $6$ .

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Step 4.6.1.1.1

Cancel the common factor.

$6 \frac{y}{6} = 6 \cdot - 3$

Step 4.6.1.1.2

Rewrite the expression.

$y = 6 \cdot - 3$

Step 4.6.2

Simplify the right side.

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Step 4.6.2.1

Multiply $6$ by $- 3$ .

$y = - 18$

Step 5

Solve $\frac{2}{3} y - 1 = - (\frac{1}{2} y - 4)$ for $y$ .

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Step 5.1

Combine $\frac{2}{3}$ and $y$ .

$\frac{2 y}{3} - 1 = - (\frac{1}{2} y - 4)$

Step 5.2

Simplify $- (\frac{1}{2} y - 4)$ .

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Step 5.2.1

Combine $\frac{1}{2}$ and $y$ .

$\frac{2 y}{3} - 1 = - (\frac{y}{2} - 4)$

Step 5.2.2

Apply the distributive property.

$\frac{2 y}{3} - 1 = - \frac{y}{2} - - 4$

Step 5.2.3

Multiply $- 1$ by $- 4$ .

$\frac{2 y}{3} - 1 = - \frac{y}{2} + 4$

Step 5.3

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

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Step 5.3.1

Add $\frac{y}{2}$ to both sides of the equation.

$\frac{2 y}{3} - 1 + \frac{y}{2} = 4$

Step 5.3.2

To write $\frac{2 y}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.3.3

To write $\frac{y}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.4.1

Multiply $\frac{2 y}{3}$ by $\frac{2}{2}$ .

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

Step 5.3.4.2

Multiply $3$ by $2$ .

$\frac{2 y \cdot 2}{6} + \frac{y}{2} \cdot \frac{3}{3} - 1 = 4$

Step 5.3.4.3

Multiply $\frac{y}{2}$ by $\frac{3}{3}$ .

$\frac{2 y \cdot 2}{6} + \frac{y \cdot 3}{2 \cdot 3} - 1 = 4$

Step 5.3.4.4

Multiply $2$ by $3$ .

$\frac{2 y \cdot 2}{6} + \frac{y \cdot 3}{6} - 1 = 4$

Step 5.3.5

Combine the numerators over the common denominator.

$\frac{2 y \cdot 2 + y \cdot 3}{6} - 1 = 4$

Step 5.3.6

Simplify each term.

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Step 5.3.6.1

Simplify the numerator.

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Step 5.3.6.1.1

Factor $y$ out of $2 y \cdot 2 + y \cdot 3$ .

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Step 5.3.6.1.1.1

Factor $y$ out of $2 y \cdot 2$ .

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

Step 5.3.6.1.1.2

Factor $y$ out of $y \cdot 3$ .

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

Step 5.3.6.1.1.3

Factor $y$ out of $y (2 \cdot 2) + y (3)$ .

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

Step 5.3.6.1.2

Multiply $2$ by $2$ .

$\frac{y (4 + 3)}{6} - 1 = 4$

Step 5.3.6.1.3

Add $4$ and $3$ .

$\frac{y \cdot 7}{6} - 1 = 4$

Step 5.3.6.2

Move $7$ to the left of $y$ .

$\frac{7 y}{6} - 1 = 4$

Step 5.4

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

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Step 5.4.1

Add $1$ to both sides of the equation.

$\frac{7 y}{6} = 4 + 1$

Step 5.4.2

Add $4$ and $1$ .

$\frac{7 y}{6} = 5$

Step 5.5

Multiply both sides of the equation by $\frac{6}{7}$ .

$\frac{6}{7} \cdot \frac{7 y}{6} = \frac{6}{7} \cdot 5$

Step 5.6

Simplify both sides of the equation.

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Step 5.6.1

Simplify the left side.

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Step 5.6.1.1

Simplify $\frac{6}{7} \cdot \frac{7 y}{6}$ .

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Step 5.6.1.1.1

Cancel the common factor of $6$ .

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Step 5.6.1.1.1.1

Cancel the common factor.

$\frac{6}{7} \cdot \frac{7 y}{6} = \frac{6}{7} \cdot 5$

Step 5.6.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 y) = \frac{6}{7} \cdot 5$

Step 5.6.1.1.2

Cancel the common factor of $7$ .

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Step 5.6.1.1.2.1

Factor $7$ out of $7 y$ .

$\frac{1}{7} (7 (y)) = \frac{6}{7} \cdot 5$

Step 5.6.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 y) = \frac{6}{7} \cdot 5$

Step 5.6.1.1.2.3

Rewrite the expression.

$y = \frac{6}{7} \cdot 5$

Step 5.6.2

Simplify the right side.

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Step 5.6.2.1

Multiply $\frac{6}{7} \cdot 5$ .

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Step 5.6.2.1.1

Combine $\frac{6}{7}$ and $5$ .

$y = \frac{6 \cdot 5}{7}$

Step 5.6.2.1.2

Multiply $6$ by $5$ .

$y = \frac{30}{7}$

Step 6

List all of the solutions.

$y = - 18, \frac{30}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = - 18, \frac{30}{7}$

Decimal Form:

$y = - 18, 4.\overline{285714}$

Mixed Number Form:

$y = - 18, 4 \frac{2}{7}$