Question
16. If \( |t+1|<8 \), then \( 7>t>-9 \). a) true b) false
Ask by Ford Beck. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The statement \(7 > t > -9\) is true for the inequality \(|t+1| < 8\).
Solution
1. We start with the inequality:
\[
|t+1| < 8
\]
2. This absolute value inequality can be rewritten as a compound inequality:
\[
-8 < t+1 < 8
\]
3. Subtract \(1\) from all three parts of the inequality:
\[
-8 - 1 < t+1 - 1 < 8 - 1
\]
which simplifies to:
\[
-9 < t < 7
\]
4. The statement given in the problem is \(7 > t > -9\), which is equivalent to:
\[
-9 < t < 7
\]
Since the two statements are equivalent, the answer is:
a) true
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Simplify this solution Extra Insights
If \( |t+1|<8 \), it means that the expression \( t+1 \) is more than -8 and less than 8. This leads us to two inequalities: \( -8 < t + 1 < 8 \). Subtracting 1 from all parts of the inequality gives us \( -9 < t < 7 \), which can be rewritten in the form \( 7 > t > -9 \). Therefore, the statement is true! So the answer is: a) true
