\( \operatorname { ion } | 5 x + 8 | = x \)

To solve the equation \( |5x + 8| = x \), we need to consider two cases due to the absolute value: when \( 5x + 8 \) is non-negative and when it is negative. For the first case, if \( 5x + 8 \geq 0 \), then \( |5x + 8| = 5x + 8 \). This gives us the equation \( 5x + 8 = x \), which simplifies to \( 4x + 8 = 0 \) or \( x = -2 \). For the second case, if \( 5x + 8 < 0 \), then \( |5x + 8| = -(5x + 8) \). This leads to \( -(5x + 8) = x \), simplifying to \( -5x - 8 = x \) or \( -6x = 8 \), giving \( x = -\frac{4}{3} \). Now we check whether these solutions satisfy the conditions of their respective cases: 1. For \( x = -2 \): \( 5(-2) + 8 = -10 + 8 = -2 \) (not valid as it doesn't satisfy \( 5x + 8 \geq 0 \)). 2. For \( x = -\frac{4}{3} \): \( 5(-\frac{4}{3}) + 8 = -\frac{20}{3} + \frac{24}{3} = \frac{4}{3} \) (valid because \( 5x + 8 \) is indeed negative). Thus the solution set is \( x = -\frac{4}{3} \). Keep in mind that when dealing with absolute value equations, always double-check your solutions against the original conditions to ensure they fit neatly!