Question - Solving Absolute Value Inequalities
mathsolving absolute value inequalitiesisolating absolute valueinequality solution stepsnumber line solution
Solution:
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.
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