Algebra Examples

$\frac{1}{2} | 5 x - 15 | = 10$

Step 1

Multiply each term in $\frac{1}{2} | 5 x - 15 | = 10$ by $2$ to eliminate the fractions.

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Step 1.1

Multiply each term in $\frac{1}{2} | 5 x - 15 | = 10$ by $2$ .

$\frac{1}{2} | 5 x - 15 | \cdot 2 = 10 \cdot 2$

Step 1.2

Simplify the left side.

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Step 1.2.1

Combine $\frac{1}{2}$ and $| 5 x - 15 |$ .

$\frac{| 5 x - 15 |}{2} \cdot 2 = 10 \cdot 2$

Step 1.2.2

Simplify the numerator.

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Step 1.2.2.1

Factor $5$ out of $5 x - 15$ .

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Step 1.2.2.1.1

Factor $5$ out of $5 x$ .

$\frac{| 5 (x) - 15 |}{2} \cdot 2 = 10 \cdot 2$

Step 1.2.2.1.2

Factor $5$ out of $- 15$ .

$\frac{| 5 x + 5 \cdot - 3 |}{2} \cdot 2 = 10 \cdot 2$

Step 1.2.2.1.3

Factor $5$ out of $5 x + 5 \cdot - 3$ .

$\frac{| 5 (x - 3) |}{2} \cdot 2 = 10 \cdot 2$

Step 1.2.2.2

Apply the distributive property.

$\frac{| 5 x + 5 \cdot - 3 |}{2} \cdot 2 = 10 \cdot 2$

Step 1.2.2.3

Multiply $5$ by $- 3$ .

$\frac{| 5 x - 15 |}{2} \cdot 2 = 10 \cdot 2$

Step 1.2.2.4

Factor $5$ out of $5 x - 15$ .

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Step 1.2.2.4.1

Factor $5$ out of $5 x$ .

$\frac{| 5 (x) - 15 |}{2} \cdot 2 = 10 \cdot 2$

Step 1.2.2.4.2

Factor $5$ out of $- 15$ .

$\frac{| 5 x + 5 \cdot - 3 |}{2} \cdot 2 = 10 \cdot 2$

Step 1.2.2.4.3

Factor $5$ out of $5 x + 5 \cdot - 3$ .

$\frac{| 5 (x - 3) |}{2} \cdot 2 = 10 \cdot 2$

Step 1.2.3

Remove non-negative terms from the absolute value.

$\frac{5 | x - 3 |}{2} \cdot 2 = 10 \cdot 2$

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

$\frac{5 | x - 3 |}{2} \cdot 2 = 10 \cdot 2$

Step 1.2.4.2

Rewrite the expression.

$5 | x - 3 | = 10 \cdot 2$

Step 1.3

Simplify the right side.

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Step 1.3.1

Multiply $10$ by $2$ .

$5 | x - 3 | = 20$

Step 2

Subtract $20$ from both sides of the equation.

$5 | x - 3 | - 20 = 0$

Step 3

Factor $5$ out of $5 | x - 3 | - 20$ .

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Step 3.1

Factor $5$ out of $5 | x - 3 |$ .

$5 (| x - 3 |) - 20 = 0$

Step 3.2

Factor $5$ out of $- 20$ .

$5 | x - 3 | + 5 \cdot - 4 = 0$

Step 3.3

Factor $5$ out of $5 | x - 3 | + 5 \cdot - 4$ .

$5 (| x - 3 | - 4) = 0$

Step 4

Divide each term in $5 (| x - 3 | - 4) = 0$ by $5$ and simplify.

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Step 4.1

Divide each term in $5 (| x - 3 | - 4) = 0$ by $5$ .

$\frac{5 (| x - 3 | - 4)}{5} = \frac{0}{5}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{5 (| x - 3 | - 4)}{5} = \frac{0}{5}$

Step 4.2.1.2

Divide $| x - 3 | - 4$ by $1$ .

$| x - 3 | - 4 = \frac{0}{5}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $5$ .

$| x - 3 | - 4 = 0$

Step 5

Add $4$ to both sides of the equation.

$| x - 3 | = 4$

Step 6

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

$x - 3 = \pm 4$

Step 7

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

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Step 7.1

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

$x - 3 = 4$

Step 7.2

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

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Step 7.2.1

Add $3$ to both sides of the equation.

$x = 4 + 3$

Step 7.2.2

Add $4$ and $3$ .

$x = 7$

Step 7.3

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

$x - 3 = - 4$

Step 7.4

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

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Step 7.4.1

Add $3$ to both sides of the equation.

$x = - 4 + 3$

Step 7.4.2

Add $- 4$ and $3$ .

$x = - 1$

Step 7.5

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

$x = 7, - 1$