Example Question - solving inequalities step by step

Solving Absolute Value Inequalities

The inequality provided in the image is \( |u + 6| \geq 46 \). Let's solve the inequality step by step. The absolute value inequality \( |u + 6| \geq 46 \) means that whatever expression is inside the absolute value signs must be either greater than or equal to 46 or less than or equal to -46. We can split this into two separate inequalities: 1. \( u + 6 \geq 46 \) 2. \( u + 6 \leq -46 \) For the first inequality, \( u + 6 \geq 46 \): Subtract 6 from both sides to get \( u \geq 40 \). For the second inequality, \( u + 6 \leq -46 \): Subtract 6 from both sides to get \( u \leq -52 \). Therefore, the solution to the inequality \( |u + 6| \geq 46 \) is \( u \geq 40 \) or \( u \leq -52 \). These solutions represent all the values of \( u \) that make the original inequality true.