Example Question - variable p

Solving Absolute Value Inequality

The equation provided in the image given is an absolute value inequality involving the variable \( p \): \[ |2p| - 8 \geq 13 \] To solve this inequality, we'll perform the following steps: 1. Isolate the absolute value expression on one side of the inequality. \[ |2p| \geq 13 + 8 \] \[ |2p| \geq 21 \] 2. Look at both scenarios which can make the absolute value expression true. Since the absolute value of an expression is the distance of that expression from zero, it can be either positive or negative. Therefore: a) \( 2p \geq 21 \) b) \( 2p \leq -21 \) 3. Solve both inequalities for \( p \): a) For \( 2p \geq 21 \): \[ p \geq \frac{21}{2} \] \[ p \geq 10.5 \] b) For \( 2p \leq -21 \): \[ p \leq \frac{-21}{2} \] \[ p \leq -10.5 \] So the solution set for the inequality \( |2p| - 8 \geq 13 \) is \( p \geq 10.5 \) or \( p \leq -10.5 \). This means that \( p \) must be less than or equal to -10.5 or greater than or equal to 10.5.