Precalculus Examples

$| 7 x - 8 | = | 9 x + 7 |$

Step 1

Rewrite the equation as $| 9 x + 7 | = | 7 x - 8 |$ .

$| 9 x + 7 | = | 7 x - 8 |$

Step 2

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

$9 x + 7 = 7 x - 8$

$9 x + 7 = - (7 x - 8)$

$- (9 x + 7) = 7 x - 8$

$- (9 x + 7) = - (7 x - 8)$

Step 3

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

$9 x + 7 = 7 x - 8$

$9 x + 7 = - (7 x - 8)$

Step 4

Solve $9 x + 7 = 7 x - 8$ 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 $7 x$ from both sides of the equation.

$9 x + 7 - 7 x = - 8$

Step 4.1.2

Subtract $7 x$ from $9 x$ .

$2 x + 7 = - 8$

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 $7$ from both sides of the equation.

$2 x = - 8 - 7$

Step 4.2.2

Subtract $7$ from $- 8$ .

$2 x = - 15$

Step 4.3

Divide each term in $2 x = - 15$ by $2$ and simplify.

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Step 4.3.1

Divide each term in $2 x = - 15$ by $2$ .

$\frac{2 x}{2} = \frac{- 15}{2}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 15}{2}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 15}{2}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Move the negative in front of the fraction.

$x = - \frac{15}{2}$

Step 5

Solve $9 x + 7 = - (7 x - 8)$ for $x$ .

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Step 5.1

Simplify $- (7 x - 8)$ .

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Step 5.1.1

Rewrite.

$9 x + 7 = 0 + 0 - (7 x - 8)$

Step 5.1.2

Simplify by adding zeros.

$9 x + 7 = - (7 x - 8)$

Step 5.1.3

Apply the distributive property.

$9 x + 7 = - (7 x) - - 8$

Step 5.1.4

Multiply.

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Step 5.1.4.1

Multiply $7$ by $- 1$ .

$9 x + 7 = - 7 x - - 8$

Step 5.1.4.2

Multiply $- 1$ by $- 8$ .

$9 x + 7 = - 7 x + 8$

Step 5.2

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

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Step 5.2.1

Add $7 x$ to both sides of the equation.

$9 x + 7 + 7 x = 8$

Step 5.2.2

Add $9 x$ and $7 x$ .

$16 x + 7 = 8$

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 $7$ from both sides of the equation.

$16 x = 8 - 7$

Step 5.3.2

Subtract $7$ from $8$ .

$16 x = 1$

Step 5.4

Divide each term in $16 x = 1$ by $16$ and simplify.

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Step 5.4.1

Divide each term in $16 x = 1$ by $16$ .

$\frac{16 x}{16} = \frac{1}{16}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $16$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{16 x}{16} = \frac{1}{16}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{16}$

Step 6

List all of the solutions.

$x = - \frac{15}{2}, \frac{1}{16}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{15}{2}, \frac{1}{16}$

Decimal Form:

$x = - 7.5, 0.0625$

Mixed Number Form:

$x = - 7 \frac{1}{2}, \frac{1}{16}$