Question
Solve the equation or inequalit \[ \begin{array}{l}\text { (a) }|b|=9 \\ \text { (b) }|b|<9 \\ \text { (c) }|b|>9\end{array} \]
Ask by Coleman Ruiz. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
(a) b = 9 or b = –9
(b) –9 < b < 9
(c) b < –9 or b > 9
Solution
We will solve each part step by step.
(a) |b| = 9
The definition of absolute value states that |b| is the distance from b to 0 on the number line. Therefore, the equation |b| = 9 means that b is 9 units away from 0. This gives us two possibilities:
b = 9 or b = –9
(b) |b| < 9
The inequality |b| < 9 means the distance of b from 0 is less than 9. Hence, b must be between –9 and 9:
–9 < b < 9
(c) |b| > 9
The inequality |b| > 9 means the distance of b from 0 is greater than 9. Therefore, b is either less than –9 or greater than 9:
b < –9 or b > 9
Thus, the solutions are:
(a) b = 9 or b = –9
(b) –9 < b < 9
(c) b < –9 or b > 9
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Simplify this solution Bonus Knowledge
For the equation \(|b|=9\), this means \(b\) can be either \(9\) or \(-9\). Essentially, the absolute value function gives us two solutions: \(b = 9\) or \(b = -9\). For the inequality \(|b|<9\), the solution tells us that \(b\) lies within the range of \(-9\) to \(9\) but does not include these endpoints. Thus, we express this as \(-9 < b < 9\). For the inequality \(|b|>9\), this implies that \(b\) is either greater than \(9\) or less than \(-9\), resulting in two intervals: \(b < -9\) or \(b > 9\). In a nutshell: (a) \(b = 9\) or \(b = -9\) (b) \(-9 < b < 9\) (c) \(b < -9\) or \(b > 9\)
