Pre-Algebra Examples

$| 3 a + 7 | = | 4 a - 5 |$

Step 1

Rewrite the equation as $| 4 a - 5 | = | 3 a + 7 |$ .

$| 4 a - 5 | = | 3 a + 7 |$

Step 2

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

$4 a - 5 = 3 a + 7$

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

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

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

Step 3

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

$4 a - 5 = 3 a + 7$

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

Step 4

Solve $4 a - 5 = 3 a + 7$ for $a$ .

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Step 4.1

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

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Step 4.1.1

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

$4 a - 5 - 3 a = 7$

Step 4.1.2

Subtract $3 a$ from $4 a$ .

$a - 5 = 7$

Step 4.2

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

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Step 4.2.1

Add $5$ to both sides of the equation.

$a = 7 + 5$

Step 4.2.2

Add $7$ and $5$ .

$a = 12$

Step 5

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

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Step 5.1

Simplify $- (3 a + 7)$ .

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Step 5.1.1

Rewrite.

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

Step 5.1.2

Simplify by adding zeros.

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

Step 5.1.3

Apply the distributive property.

$4 a - 5 = - (3 a) - 1 \cdot 7$

Step 5.1.4

Multiply.

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Step 5.1.4.1

Multiply $3$ by $- 1$ .

$4 a - 5 = - 3 a - 1 \cdot 7$

Step 5.1.4.2

Multiply $- 1$ by $7$ .

$4 a - 5 = - 3 a - 7$

Step 5.2

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

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Step 5.2.1

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

$4 a - 5 + 3 a = - 7$

Step 5.2.2

Add $4 a$ and $3 a$ .

$7 a - 5 = - 7$

Step 5.3

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

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Step 5.3.1

Add $5$ to both sides of the equation.

$7 a = - 7 + 5$

Step 5.3.2

Add $- 7$ and $5$ .

$7 a = - 2$

Step 5.4

Divide each term in $7 a = - 2$ by $7$ and simplify.

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Step 5.4.1

Divide each term in $7 a = - 2$ by $7$ .

$\frac{7 a}{7} = \frac{- 2}{7}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $7$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{7 a}{7} = \frac{- 2}{7}$

Step 5.4.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 2}{7}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Move the negative in front of the fraction.

$a = - \frac{2}{7}$

Step 6

List all of the solutions.

$a = 12, - \frac{2}{7}$