Pre-Algebra Examples

$| 3 (x - 4) + 7 | = | 3 x - 5 |$

Step 1

Rewrite the equation as $| 3 x - 5 | = | 3 (x - 4) + 7 |$ .

$| 3 x - 5 | = | 3 (x - 4) + 7 |$

Step 2

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

$3 x - 5 = 3 (x - 4) + 7$

$3 x - 5 = - (3 (x - 4) + 7)$

$- (3 x - 5) = 3 (x - 4) + 7$

$- (3 x - 5) = - (3 (x - 4) + 7)$

Step 3

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

$3 x - 5 = 3 (x - 4) + 7$

$3 x - 5 = - (3 (x - 4) + 7)$

Step 4

Solve $3 x - 5 = 3 (x - 4) + 7$ for $x$ .

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Step 4.1

Simplify $3 (x - 4) + 7$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the distributive property.

$3 x - 5 = 3 x + 3 \cdot - 4 + 7$

Step 4.1.1.2

Multiply $3$ by $- 4$ .

$3 x - 5 = 3 x - 12 + 7$

Step 4.1.2

Add $- 12$ and $7$ .

$3 x - 5 = 3 x - 5$

Step 4.2

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

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Step 4.2.1

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

$3 x - 5 - 3 x = - 5$

Step 4.2.2

Combine the opposite terms in $3 x - 5 - 3 x$ .

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Step 4.2.2.1

Subtract $3 x$ from $3 x$ .

$0 - 5 = - 5$

Step 4.2.2.2

Subtract $5$ from $0$ .

$- 5 = - 5$

Step 4.3

Since $- 5 = - 5$ , the equation will always be true.

All real numbers

Step 5

Solve $3 x - 5 = - (3 (x - 4) + 7)$ for $x$ .

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Step 5.1

Simplify $- (3 (x - 4) + 7)$ .

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Step 5.1.1

Rewrite.

$3 x - 5 = 0 + 0 - (3 (x - 4) + 7)$

Step 5.1.2

Simplify by adding zeros.

$3 x - 5 = - (3 (x - 4) + 7)$

Step 5.1.3

Simplify each term.

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Step 5.1.3.1

Apply the distributive property.

$3 x - 5 = - (3 x + 3 \cdot - 4 + 7)$

Step 5.1.3.2

Multiply $3$ by $- 4$ .

$3 x - 5 = - (3 x - 12 + 7)$

Step 5.1.4

Simplify by multiplying through.

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Step 5.1.4.1

Add $- 12$ and $7$ .

$3 x - 5 = - (3 x - 5)$

Step 5.1.4.2

Apply the distributive property.

$3 x - 5 = - (3 x) - - 5$

Step 5.1.4.3

Multiply.

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Step 5.1.4.3.1

Multiply $3$ by $- 1$ .

$3 x - 5 = - 3 x - - 5$

Step 5.1.4.3.2

Multiply $- 1$ by $- 5$ .

$3 x - 5 = - 3 x + 5$

Step 5.2

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

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Step 5.2.1

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

$3 x - 5 + 3 x = 5$

Step 5.2.2

Add $3 x$ and $3 x$ .

$6 x - 5 = 5$

Step 5.3

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

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Step 5.3.1

Add $5$ to both sides of the equation.

$6 x = 5 + 5$

Step 5.3.2

Add $5$ and $5$ .

$6 x = 10$

Step 5.4

Divide each term in $6 x = 10$ by $6$ and simplify.

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Step 5.4.1

Divide each term in $6 x = 10$ by $6$ .

$\frac{6 x}{6} = \frac{10}{6}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $6$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{10}{6}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{6}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Cancel the common factor of $10$ and $6$ .

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Step 5.4.3.1.1

Factor $2$ out of $10$ .

$x = \frac{2 (5)}{6}$

Step 5.4.3.1.2

Cancel the common factors.

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Step 5.4.3.1.2.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot 5}{2 \cdot 3}$

Step 5.4.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 5}{2 \cdot 3}$

Step 5.4.3.1.2.3

Rewrite the expression.

$x = \frac{5}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{3}$

Decimal Form:

$x = 1.\overline{6}$

Mixed Number Form:

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