Question - Solving Absolute Value Inequalities
mathsolving inequalitiesabsolute value inequalitiesinequality solutionsisolating absolute value expression
Solution:
Sure, let's solve the inequality presented in the image:\[ |2p| - 8 \geq 13 \]To solve this, you first need to isolate the absolute value expression on one side of the inequality. Let's move the -8 to the other side by adding 8 to both sides:\[ |2p| \geq 13 + 8 \]\[ |2p| \geq 21 \]Now that the absolute value is isolated, we can split the inequality into two separate cases, because the absolute value of a number is either equal to the number itself if the number is positive or equal to the negative of the number if it is negative. So we have:\[ 2p \geq 21 \quad \text{or} \quad 2p \leq -21 \]Now, solve each inequality for $$ p $$:For the first inequality:\[ 2p \geq 21 \]\[ p \geq \frac{21}{2} \]\[ p \geq 10.5 \]For the second inequality:\[ 2p \leq -21 \]\[ p \leq \frac{-21}{2} \]\[ p \leq -10.5 \]Therefore, the solution to the inequality is $$ p \leq -10.5 $$ or $$ p \geq 10.5 $$. This means $$ p $$ can be any number less than or equal to -10.5 or any number greater than or equal to 10.5.
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