Calculus Examples

$| 10 x - 12 | = | 7 x + 9 |$

Step 1

Rewrite the equation as $| 7 x + 9 | = | 10 x - 12 |$ .

$| 7 x + 9 | = | 10 x - 12 |$

Step 2

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

$7 x + 9 = 10 x - 12$

$7 x + 9 = - (10 x - 12)$

$- (7 x + 9) = 10 x - 12$

$- (7 x + 9) = - (10 x - 12)$

Step 3

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

$7 x + 9 = 10 x - 12$

$7 x + 9 = - (10 x - 12)$

Step 4

Solve $7 x + 9 = 10 x - 12$ for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $10 x$ from both sides of the equation.

$7 x + 9 - 10 x = - 12$

Step 4.1.2

Subtract $10 x$ from $7 x$ .

$- 3 x + 9 = - 12$

Step 4.2

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

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Step 4.2.1

Subtract $9$ from both sides of the equation.

$- 3 x = - 12 - 9$

Step 4.2.2

Subtract $9$ from $- 12$ .

$- 3 x = - 21$

Step 4.3

Divide each term in $- 3 x = - 21$ by $- 3$ and simplify.

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Step 4.3.1

Divide each term in $- 3 x = - 21$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 21}{- 3}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 3$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 21}{- 3}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 21}{- 3}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $- 21$ by $- 3$ .

$x = 7$

Step 5

Solve $7 x + 9 = - (10 x - 12)$ for $x$ .

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Step 5.1

Simplify $- (10 x - 12)$ .

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Step 5.1.1

Rewrite.

$7 x + 9 = 0 + 0 - (10 x - 12)$

Step 5.1.2

Simplify by adding zeros.

$7 x + 9 = - (10 x - 12)$

Step 5.1.3

Apply the distributive property.

$7 x + 9 = - (10 x) - - 12$

Step 5.1.4

Multiply.

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Step 5.1.4.1

Multiply $10$ by $- 1$ .

$7 x + 9 = - 10 x - - 12$

Step 5.1.4.2

Multiply $- 1$ by $- 12$ .

$7 x + 9 = - 10 x + 12$

Step 5.2

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

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Step 5.2.1

Add $10 x$ to both sides of the equation.

$7 x + 9 + 10 x = 12$

Step 5.2.2

Add $7 x$ and $10 x$ .

$17 x + 9 = 12$

Step 5.3

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

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Step 5.3.1

Subtract $9$ from both sides of the equation.

$17 x = 12 - 9$

Step 5.3.2

Subtract $9$ from $12$ .

$17 x = 3$

Step 5.4

Divide each term in $17 x = 3$ by $17$ and simplify.

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Step 5.4.1

Divide each term in $17 x = 3$ by $17$ .

$\frac{17 x}{17} = \frac{3}{17}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $17$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{17 x}{17} = \frac{3}{17}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{17}$

Step 6

List all of the solutions.

$x = 7, \frac{3}{17}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 7, \frac{3}{17}$

Decimal Form:

$x = 7, 0.17647058 \dots$