Algebra Examples

$\frac{3}{4} | 8 t - 12 | = 6 (t - 1)$

Step 1

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} (\frac{3}{4} | 8 t - 12 |) = \frac{4}{3} (6 (t - 1))$

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 $\frac{4}{3} (\frac{3}{4} | 8 t - 12 |)$ .

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Step 2.1.1.1

Combine $\frac{3}{4}$ and $| 8 t - 12 |$ .

$\frac{4}{3} \cdot \frac{3 | 8 t - 12 |}{4} = \frac{4}{3} (6 (t - 1))$

Step 2.1.1.2

Cancel the common factor of $4$ .

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Step 2.1.1.2.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 | 8 t - 12 |}{4} = \frac{4}{3} (6 (t - 1))$

Step 2.1.1.2.2

Rewrite the expression.

$\frac{1}{3} (3 | 8 t - 12 |) = \frac{4}{3} (6 (t - 1))$

Step 2.1.1.3

Cancel the common factor of $3$ .

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Step 2.1.1.3.1

Factor $3$ out of $3 | 8 t - 12 |$ .

$\frac{1}{3} (3 | 8 t - 12 |) = \frac{4}{3} (6 (t - 1))$

Step 2.1.1.3.2

Cancel the common factor.

$\frac{1}{3} (3 | 8 t - 12 |) = \frac{4}{3} (6 (t - 1))$

Step 2.1.1.3.3

Rewrite the expression.

$| 8 t - 12 | = \frac{4}{3} (6 (t - 1))$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{4}{3} (6 (t - 1))$ .

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Step 2.2.1.1

Cancel the common factor of $3$ .

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Step 2.2.1.1.1

Factor $3$ out of $6 (t - 1)$ .

$| 8 t - 12 | = \frac{4}{3} (3 (2 (t - 1)))$

Step 2.2.1.1.2

Cancel the common factor.

$| 8 t - 12 | = \frac{4}{3} (3 (2 (t - 1)))$

Step 2.2.1.1.3

Rewrite the expression.

$| 8 t - 12 | = 4 (2 (t - 1))$

Step 2.2.1.2

Multiply $2$ by $4$ .

$| 8 t - 12 | = 8 (t - 1)$

Step 2.2.1.3

Apply the distributive property.

$| 8 t - 12 | = 8 t + 8 \cdot - 1$

Step 2.2.1.4

Multiply $8$ by $- 1$ .

$| 8 t - 12 | = 8 t - 8$

Step 3

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

$8 t - 12 = \pm (8 t - 8)$

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.

$8 t - 12 = 8 t - 8$

Step 4.2

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

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Step 4.2.1

Subtract $8 t$ from both sides of the equation.

$8 t - 12 - 8 t = - 8$

Step 4.2.2

Combine the opposite terms in $8 t - 12 - 8 t$ .

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Step 4.2.2.1

Subtract $8 t$ from $8 t$ .

$0 - 12 = - 8$

Step 4.2.2.2

Subtract $12$ from $0$ .

$- 12 = - 8$

Step 4.3

Since $- 12 \neq - 8$ , there are no solutions.

No solution

Step 4.4

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

$8 t - 12 = - (8 t - 8)$

Step 4.5

Simplify $- (8 t - 8)$ .

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Step 4.5.1

Rewrite.

$8 t - 12 = 0 + 0 - (8 t - 8)$

Step 4.5.2

Simplify by adding zeros.

$8 t - 12 = - (8 t - 8)$

Step 4.5.3

Apply the distributive property.

$8 t - 12 = - (8 t) - - 8$

Step 4.5.4

Multiply.

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Step 4.5.4.1

Multiply $8$ by $- 1$ .

$8 t - 12 = - 8 t - - 8$

Step 4.5.4.2

Multiply $- 1$ by $- 8$ .

$8 t - 12 = - 8 t + 8$

Step 4.6

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

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Step 4.6.1

Add $8 t$ to both sides of the equation.

$8 t - 12 + 8 t = 8$

Step 4.6.2

Add $8 t$ and $8 t$ .

$16 t - 12 = 8$

Step 4.7

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

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Step 4.7.1

Add $12$ to both sides of the equation.

$16 t = 8 + 12$

Step 4.7.2

Add $8$ and $12$ .

$16 t = 20$

Step 4.8

Divide each term in $16 t = 20$ by $16$ and simplify.

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Step 4.8.1

Divide each term in $16 t = 20$ by $16$ .

$\frac{16 t}{16} = \frac{20}{16}$

Step 4.8.2

Simplify the left side.

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Step 4.8.2.1

Cancel the common factor of $16$ .

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Step 4.8.2.1.1

Cancel the common factor.

$\frac{16 t}{16} = \frac{20}{16}$

Step 4.8.2.1.2

Divide $t$ by $1$ .

$t = \frac{20}{16}$

Step 4.8.3

Simplify the right side.

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Step 4.8.3.1

Cancel the common factor of $20$ and $16$ .

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Step 4.8.3.1.1

Factor $4$ out of $20$ .

$t = \frac{4 (5)}{16}$

Step 4.8.3.1.2

Cancel the common factors.

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Step 4.8.3.1.2.1

Factor $4$ out of $16$ .

$t = \frac{4 \cdot 5}{4 \cdot 4}$

Step 4.8.3.1.2.2

Cancel the common factor.

$t = \frac{4 \cdot 5}{4 \cdot 4}$

Step 4.8.3.1.2.3

Rewrite the expression.

$t = \frac{5}{4}$

Step 4.9

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

$t =$