Basic Math Examples

$\frac{1}{3} \cdot | 4 y - 11 | = y + 4$

Step 1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} | 4 y - 11 |) = 3 (y + 4)$

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $3 (\frac{1}{3} | 4 y - 11 |)$ .

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Step 2.1.1.1

Combine $\frac{1}{3}$ and $| 4 y - 11 |$ .

$3 \frac{| 4 y - 11 |}{3} = 3 (y + 4)$

Step 2.1.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.2.1

Cancel the common factor.

$3 \frac{| 4 y - 11 |}{3} = 3 (y + 4)$

Step 2.1.1.2.2

Rewrite the expression.

$| 4 y - 11 | = 3 (y + 4)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $3 (y + 4)$ .

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Step 2.2.1.1

Apply the distributive property.

$| 4 y - 11 | = 3 y + 3 \cdot 4$

Step 2.2.1.2

Multiply $3$ by $4$ .

$| 4 y - 11 | = 3 y + 12$

Step 3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$4 y - 11 = \pm (3 y + 12)$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$4 y - 11 = 3 y + 12$

Step 4.2

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

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Step 4.2.1

Subtract $3 y$ from both sides of the equation.

$4 y - 11 - 3 y = 12$

Step 4.2.2

Subtract $3 y$ from $4 y$ .

$y - 11 = 12$

Step 4.3

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

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Step 4.3.1

Add $11$ to both sides of the equation.

$y = 12 + 11$

Step 4.3.2

Add $12$ and $11$ .

$y = 23$

Step 4.4

Next, use the negative value of the $\pm$ to find the second solution.

$4 y - 11 = - (3 y + 12)$

Step 4.5

Simplify $- (3 y + 12)$ .

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Step 4.5.1

Rewrite.

$4 y - 11 = 0 + 0 - (3 y + 12)$

Step 4.5.2

Simplify by adding zeros.

$4 y - 11 = - (3 y + 12)$

Step 4.5.3

Apply the distributive property.

$4 y - 11 = - (3 y) - 1 \cdot 12$

Step 4.5.4

Multiply.

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Step 4.5.4.1

Multiply $3$ by $- 1$ .

$4 y - 11 = - 3 y - 1 \cdot 12$

Step 4.5.4.2

Multiply $- 1$ by $12$ .

$4 y - 11 = - 3 y - 12$

Step 4.6

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

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Step 4.6.1

Add $3 y$ to both sides of the equation.

$4 y - 11 + 3 y = - 12$

Step 4.6.2

Add $4 y$ and $3 y$ .

$7 y - 11 = - 12$

Step 4.7

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

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Step 4.7.1

Add $11$ to both sides of the equation.

$7 y = - 12 + 11$

Step 4.7.2

Add $- 12$ and $11$ .

$7 y = - 1$

Step 4.8

Divide each term in $7 y = - 1$ by $7$ and simplify.

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Step 4.8.1

Divide each term in $7 y = - 1$ by $7$ .

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

Step 4.8.2

Simplify the left side.

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Step 4.8.2.1

Cancel the common factor of $7$ .

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Step 4.8.2.1.1

Cancel the common factor.

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

Step 4.8.2.1.2

Divide $y$ by $1$ .

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

Step 4.8.3

Simplify the right side.

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Step 4.8.3.1

Move the negative in front of the fraction.

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

Step 4.9

The complete solution is the result of both the positive and negative portions of the solution.

$y = 23, - \frac{1}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = 23, - \frac{1}{7}$

Decimal Form:

$y = 23, - 0.\overline{142857}$