Solving Absolute Value Inequalities
To solve the inequality \( 9 - 8 | r + 5 | > -11 \), we must isolate the absolute value expression on one side of the inequality.
Here are the steps to solve the inequality:
1. Add or subtract quantities from both sides of the inequality to move constant terms away from the absolute value expression:
\[ 9 - 8 | r + 5 | + 11 > -11 + 11 \]
\[ 20 - 8 | r + 5 | > 0 \]
2. Now we need to isolate the absolute value expression. To do this, we can subtract 20 from both sides:
\[ 20 - 8 | r + 5 | - 20 > 0 - 20 \]
\[ -8 | r + 5 | > -20 \]
3. To get rid of the coefficient in front of the absolute value, we divide both sides by -8. Remember that when you divide by a negative number, you have to flip the inequality sign:
\[ \frac{-8 | r + 5 |}{-8} < \frac{-20}{-8} \]
\[ | r + 5 | < \frac{20}{8} \]
4. Simplify the fraction:
\[ | r + 5 | < 2.5 \]
This inequality says that the distance from the point \( r + 5 \) to 0 on the number line is less than 2.5. We can break this down into two separate inequalities:
If \( r + 5 \) is positive:
\[ r + 5 < 2.5 \]
If \( r + 5 \) is negative:
\[ r + 5 > -2.5 \]
Now solve for \( r \) in each inequality:
For the case where \( r + 5 \) is positive:
\[ r < 2.5 - 5 \]
\[ r < -2.5 \]
For the case where \( r + 5 \) is negative:
\[ r > -2.5 - 5 \]
\[ r > -7.5 \]
So the solution to the inequality is:
\[ -7.5 < r < -2.5 \]
This means that \( r \) is greater than -7.5 and less than -2.5.
