Solve the Absolute Value Inequality for w |5w+8|=|5w+7|

$| 5 w + 8 | = | 5 w + 7 |$

Rewrite the equation as $| 5 w + 7 | = | 5 w + 8 |$ .

$| 5 w + 7 | = | 5 w + 8 |$

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

$5 w + 7 = 5 w + 8$

$5 w + 7 = - (5 w + 8)$

$- (5 w + 7) = 5 w + 8$

$- (5 w + 7) = - (5 w + 8)$

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

$5 w + 7 = 5 w + 8$

$5 w + 7 = - (5 w + 8)$

Solve $5 w + 7 = 5 w + 8$ for $w$ .

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

Subtract $5 w$ from both sides of the equation.

$5 w + 7 - 5 w = 8$

Combine the opposite terms in $5 w + 7 - 5 w$ .

Subtract $5 w$ from $5 w$ .

$0 + 7 = 8$

Add $0$ and $7$ .

$7 = 8$

Since $7 \neq 8$ , there are no solutions.

No solution

Solve $5 w + 7 = - (5 w + 8)$ for $w$ .

Simplify $- (5 w + 8)$ .

Rewrite.

$5 w + 7 = 0 + 0 - (5 w + 8)$

Simplify by adding zeros.

$5 w + 7 = - (5 w + 8)$

Apply the distributive property.

$5 w + 7 = - (5 w) - 1 \cdot 8$

Multiply.

Multiply $5$ by $- 1$ .

$5 w + 7 = - 5 w - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$5 w + 7 = - 5 w - 8$

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

Add $5 w$ to both sides of the equation.

$5 w + 7 + 5 w = - 8$

Add $5 w$ and $5 w$ .

$10 w + 7 = - 8$

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

Subtract $7$ from both sides of the equation.

$10 w = - 8 - 7$

Subtract $7$ from $- 8$ .

$10 w = - 15$

Divide each term in $10 w = - 15$ by $10$ and simplify.

Divide each term in $10 w = - 15$ by $10$ .

$\frac{10 w}{10} = \frac{- 15}{10}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 w}{10} = \frac{- 15}{10}$

Divide $w$ by $1$ .

$w = \frac{- 15}{10}$

Simplify the right side.

Cancel the common factor of $- 15$ and $10$ .

Factor $5$ out of $- 15$ .

$w = \frac{5 (- 3)}{10}$

Cancel the common factors.

Factor $5$ out of $10$ .

$w = \frac{5 \cdot - 3}{5 \cdot 2}$

Cancel the common factor.

$w = \frac{5 \cdot - 3}{5 \cdot 2}$

Rewrite the expression.

$w = \frac{- 3}{2}$

Move the negative in front of the fraction.

$w = - \frac{3}{2}$

List all of the solutions.

$w = - \frac{3}{2}$

Verify each of the solutions by substituting them into $| 5 w + 8 | = | 5 w + 7 |$ and solving.

$w = - \frac{3}{2}$

The result can be shown in multiple forms.

Exact Form:

$w = - \frac{3}{2}$

Decimal Form:

$w = - 1.5$

Mixed Number Form:

$w = - 1 \frac{1}{2}$

Solve the Absolute Value Inequality for w |5w+8|=|5w+7|

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