Question - Solving Absolute Value Inequalities
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Solution:
To solve the inequality $$ |6 - w| < 59 $$, we approach it by considering the definition of the absolute value.The absolute value $$ |a| $$ of a number is its distance from zero on the number line, regardless of direction.This means $$ |6 - w| < 59 $$ tells us that the expression $$ 6 - w $$ must be less than 59 units away from 0 on the number line.We must consider two cases due to the absolute value:1. $$ 6 - w $$ is positive or zero:\[ 6 - w < 59 \] \[ -w < 59 - 6 \] \[ -w < 53 \] \[ w > -53 \](When we divide or multiply by a negative number, we reverse the inequality sign).2. $$ 6 - w $$ is negative:\[ -(6 - w) < 59 \] \[ w - 6 < 59 \] \[ w < 59 + 6 \] \[ w < 65 \]Therefore, combining both cases, the solution to the inequality is:\[ -53 < w < 65 \]This tells us w must be greater than -53 and less than 65.
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