Question - Solving Inequalities with Absolute Value
Solution:
To solve the inequality $$ |t - 75| \leq 15 $$, we need to split it into two separate inequalities, because the absolute value of a number is the distance from zero on the number line, and it can be expressed as the value being either greater than or equal to 0 or less than or equal to 0.The two cases for $$ |t - 75| \leq 15 $$ are:1. $$ t - 75 \leq 15 $$2. $$ -(t - 75) \leq 15 $$ or equivalently $$ t - 75 \geq -15 $$For the first case (1):$$ t - 75 \leq 15 $$Adding 75 to both sides of the inequality gives us:$$ t \leq 90 $$For the second case (2):$$ t - 75 \geq -15 $$Adding 75 to both sides of this inequality gives us:$$ t \geq 60 $$So, the solution to $$ |t - 75| \leq 15 $$ is the set of values of $$ t $$ that satisfy both conditions:$$ 60 \leq t \leq 90 $$This means that the value of $$ t $$ is between 60 and 90, inclusive.
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