Solving Absolute Value Inequalities
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.
