Question - Solving Absolute Value Inequalities
mathsolving absolute value inequalitiesabsolute value expressioninequality with absolute valuepositive and negative solutions
Solution:
To solve the inequality $$ |u - 7| > 29 $$, we need to split it into two separate inequalities because the absolute value expression can represent a positive or negative result:1. If the expression inside the absolute value, $$ u - 7 $$, is positive, then $$ u - 7 > 29 $$. To solve for $$ u $$, add 7 to both sides of the inequality:$$ u - 7 + 7 > 29 + 7 $$$$ u > 36 $$2. If the expression inside the absolute value, $$ u - 7 $$, is negative, then $$ -(u - 7) > 29 $$. This is equivalent to $$ 7 - u > 29 $$. To solve for $$ u $$, first subtract 7 from both sides:$$ 7 - u - 7 > 29 - 7 $$$$ -u > 22 $$Next, multiply both sides by -1 to isolate $$ u $$, and remember to reverse the inequality sign when multiplying or dividing by a negative number:$$ -u(-1) < 22(-1) $$$$ u < -22 $$Combine both solutions to express the full solution to the original inequality:$$ u > 36 $$ or $$ u < -22 $$Hence, $$ u $$ is any number greater than 36 or any number less than -22.
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