Algebra Examples

$4 | 1 - \frac{3}{4} x | + 7 = 10$

Step 1

Move all terms not containing $| 1 - \frac{3}{4} x |$ to the right side of the equation.

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Step 1.1

Subtract $7$ from both sides of the equation.

$4 | 1 - \frac{3}{4} x | = 10 - 7$

Step 1.2

Subtract $7$ from $10$ .

$4 | 1 - \frac{3}{4} x | = 3$

Step 2

Divide each term in $4 | 1 - \frac{3}{4} x | = 3$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 | 1 - \frac{3}{4} x | = 3$ by $4$ .

$\frac{4 | 1 - \frac{3}{4} x |}{4} = \frac{3}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 | 1 - \frac{3}{4} x |}{4} = \frac{3}{4}$

Step 2.2.1.2

Divide $| 1 - \frac{3}{4} x |$ by $1$ .

$| 1 - \frac{3}{4} x | = \frac{3}{4}$

Step 3

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

$1 - \frac{3}{4} x = \pm \frac{3}{4}$

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.

$1 - \frac{3}{4} x = \frac{3}{4}$

Step 4.2

Simplify each term.

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Step 4.2.1

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

$1 - \frac{x \cdot 3}{4} = \frac{3}{4}$

Step 4.2.2

Move $3$ to the left of $x$ .

$1 - \frac{3 x}{4} = \frac{3}{4}$

Step 4.3

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

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Step 4.3.1

Subtract $1$ from both sides of the equation.

$- \frac{3 x}{4} = \frac{3}{4} - 1$

Step 4.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{3 x}{4} = \frac{3}{4} - 1 \cdot \frac{4}{4}$

Step 4.3.3

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

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

Step 4.3.4

Combine the numerators over the common denominator.

$- \frac{3 x}{4} = \frac{3 - 1 \cdot 4}{4}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $- 1$ by $4$ .

$- \frac{3 x}{4} = \frac{3 - 4}{4}$

Step 4.3.5.2

Subtract $4$ from $3$ .

$- \frac{3 x}{4} = \frac{- 1}{4}$

Step 4.3.6

Move the negative in front of the fraction.

$- \frac{3 x}{4} = - \frac{1}{4}$

Step 4.4

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

$- 3 x = - 1$

Step 4.5

Divide each term in $- 3 x = - 1$ by $- 3$ and simplify.

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Step 4.5.1

Divide each term in $- 3 x = - 1$ by $- 3$ .

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

Step 4.5.2

Simplify the left side.

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Step 4.5.2.1

Cancel the common factor of $- 3$ .

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Step 4.5.2.1.1

Cancel the common factor.

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

Step 4.5.2.1.2

Divide $x$ by $1$ .

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

Step 4.5.3

Simplify the right side.

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Step 4.5.3.1

Dividing two negative values results in a positive value.

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

Step 4.6

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

$1 - \frac{3}{4} x = - \frac{3}{4}$

Step 4.7

Simplify each term.

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Step 4.7.1

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

$1 - \frac{x \cdot 3}{4} = - \frac{3}{4}$

Step 4.7.2

Move $3$ to the left of $x$ .

$1 - \frac{3 x}{4} = - \frac{3}{4}$

Step 4.8

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

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Step 4.8.1

Subtract $1$ from both sides of the equation.

$- \frac{3 x}{4} = - \frac{3}{4} - 1$

Step 4.8.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{3 x}{4} = - \frac{3}{4} - 1 \cdot \frac{4}{4}$

Step 4.8.3

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

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

Step 4.8.4

Combine the numerators over the common denominator.

$- \frac{3 x}{4} = \frac{- 3 - 1 \cdot 4}{4}$

Step 4.8.5

Simplify the numerator.

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Step 4.8.5.1

Multiply $- 1$ by $4$ .

$- \frac{3 x}{4} = \frac{- 3 - 4}{4}$

Step 4.8.5.2

Subtract $4$ from $- 3$ .

$- \frac{3 x}{4} = \frac{- 7}{4}$

Step 4.8.6

Move the negative in front of the fraction.

$- \frac{3 x}{4} = - \frac{7}{4}$

Step 4.9

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

$- 3 x = - 7$

Step 4.10

Divide each term in $- 3 x = - 7$ by $- 3$ and simplify.

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Step 4.10.1

Divide each term in $- 3 x = - 7$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 7}{- 3}$

Step 4.10.2

Simplify the left side.

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Step 4.10.2.1

Cancel the common factor of $- 3$ .

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Step 4.10.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 7}{- 3}$

Step 4.10.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{- 3}$

Step 4.10.3

Simplify the right side.

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Step 4.10.3.1

Dividing two negative values results in a positive value.

$x = \frac{7}{3}$

Step 4.11

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

$x = \frac{1}{3}, \frac{7}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}, \frac{7}{3}$

Decimal Form:

$x = 0.\overline{3}, 2.\overline{3}$

Mixed Number Form:

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